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Question 22170: Keep mu(d) Paired with F(n Divided by d)

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0043-22170 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0045

Original question and givens

Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=5, b=7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

  • G001 — Use positive integer n=20, exact integer arithmetic function f(k)=a*k+b with a=5, b=7. F(m) sums f(d) over every positive divisor d of m. Inversion uses mu(d)*F(m/d); complete prime multiplicities determine mu, with mu(1)=1. Do not replace d|m by d≤m or use only prime/proper divisors. Give (n, inverse_terms, reconstructed, f_n, equal).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source operation=mobinv-inverse-reconstruction;original snapshot and mathematics retained

After

explicit ordered source output fields=[n, inverse_terms, reconstructed, f_n, equal]

Conditions: none

Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

Mobius inversion ORIGINAL inverse:actual positive input=20;check complete integer range1..20

After

ALL positive divisors INCREASING=[1, 2, 4, 5, 10, 20];not only primes/proper divisors or all integers<=n

Conditions: none

Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.

Step identifiers and dependencies

Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S001

S003 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:actual positive input for mu=1

After

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Conditions: none

Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.

Step identifiers and dependencies

Rule: mobius_complete_UPF_input_role · Uses: G001, S002

S004 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor1:input=1;COMPLETE prime multiset=[];distinct primes=[]

After

mu=1;empty product case=Yes;repeated prime=No

Conditions: none

The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.

Step identifiers and dependencies

Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S003

S005 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 := div(20, 1)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 20

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

Mobius inversion ORIGINAL inverse F(n/1):actual positive input=20;check complete integer range1..20

After

ALL positive divisors INCREASING=[1, 2, 4, 5, 10, 20];not only primes/proper divisors or all integers<=n

Conditions: none

Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.

Step identifiers and dependencies

Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S005

S007 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor := mul(5, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 5

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b := sum(5, 7)

After

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 12

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor := mul(5, 2)

After

Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor = 10

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b := sum(10, 7)

After

Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b = 17

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor := mul(5, 4)

After

Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor = 20

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b := sum(20, 7)

After

Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b = 27

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor := mul(5, 5)

After

Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor = 25

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b := sum(25, 7)

After

Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b = 32

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor := mul(5, 10)

After

Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor = 50

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b := sum(50, 7)

After

Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b = 57

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(20) ORIGINAL a times divisor := mul(5, 20)

After

Mobius inversion ORIGINAL inverse F(n/1) f(20) ORIGINAL a times divisor = 100

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) f(20) plus ORIGINAL b := sum(100, 7)

After

Mobius inversion ORIGINAL inverse F(n/1) f(20) plus ORIGINAL b = 107

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum := sum(12, 17, 27, 32, 57, 107)

After

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 252

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) := mul(1, 252)

After

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 252

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:actual positive input for mu=2

After

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Conditions: none

Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.

Step identifiers and dependencies

Rule: mobius_complete_UPF_input_role · Uses: G001, S020

S022 · implies

Before

ORIGINAL positive integer absolute magnitude=2>1

After

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

Conditions: none

Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.

Step identifiers and dependencies

Rule: upf_factorization_remaining_product_invariant · Uses: G001, S021

S023 · calculation

Before

upf canonical complete product factor0 := mul(1, 2)

After

upf canonical complete product factor0 = 2

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

After

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Conditions: none

If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.

Step identifiers and dependencies

Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S023

S025 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor2:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

After

mu=-1;empty product case=No;repeated prime=No

Conditions: none

The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.

Step identifiers and dependencies

Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S024

S026 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 := div(20, 2)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 = 10

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

Mobius inversion ORIGINAL inverse F(n/2):actual positive input=10;check complete integer range1..10

After

ALL positive divisors INCREASING=[1, 2, 5, 10];not only primes/proper divisors or all integers<=n

Conditions: none

Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.

Step identifiers and dependencies

Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S026

S028 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor := mul(5, 1)

After

Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor = 5

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b := sum(5, 7)

After

Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b = 12

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor := mul(5, 2)

After

Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor = 10

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b := sum(10, 7)

After

Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b = 17

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor := mul(5, 5)

After

Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor = 25

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b := sum(25, 7)

After

Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b = 32

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(10) ORIGINAL a times divisor := mul(5, 10)

After

Mobius inversion ORIGINAL inverse F(n/2) f(10) ORIGINAL a times divisor = 50

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) f(10) plus ORIGINAL b := sum(50, 7)

After

Mobius inversion ORIGINAL inverse F(n/2) f(10) plus ORIGINAL b = 57

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum := sum(12, 17, 32, 57)

After

Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum = 118

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) := mul(-1, 118)

After

Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) = -118

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:actual positive input for mu=4

After

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Conditions: none

Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.

Step identifiers and dependencies

Rule: mobius_complete_UPF_input_role · Uses: G001, S037

S039 · implies

Before

ORIGINAL positive integer absolute magnitude=4>1

After

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

Conditions: none

Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.

Step identifiers and dependencies

Rule: upf_factorization_remaining_product_invariant · Uses: G001, S038

S040 · calculation

Before

upf extract divisor2 occurrence0 := div(4, 2)

After

upf extract divisor2 occurrence0 = 2

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

upf extract divisor2 occurrence1 := div(2, 2)

After

upf extract divisor2 occurrence1 = 1

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · implies

Before

next divisor=2;previous remaining=4;all smaller divisors exhausted

After

extracted multiplicity=2;next remaining=1;remaining modulo2=1;extracted multiset=[2, 2]

Conditions: none

Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.

Step identifiers and dependencies

Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S041

S043 · calculation

Before

upf canonical complete product factor0 := mul(1, 2)

After

upf canonical complete product factor0 = 2

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

upf canonical complete product factor1 := mul(2, 2)

After

upf canonical complete product factor1 = 4

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · implies

Before

exhausted trial divisors below3;terminal remaining=1;d*d>remaining;ORIGINAL n=4

After

complete increasing prime multiset=[2, 2];full product=4;a composite terminal remainder would have an already tested smaller divisor

Conditions: none

If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.

Step identifiers and dependencies

Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S044

S046 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor4:input=4;COMPLETE prime multiset=[2, 2];distinct primes=[2]

After

mu=0;empty product case=No;repeated prime=Yes

Conditions: none

The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.

Step identifiers and dependencies

Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S045

S047 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 := div(20, 4)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 = 5

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

Mobius inversion ORIGINAL inverse F(n/4):actual positive input=5;check complete integer range1..5

After

ALL positive divisors INCREASING=[1, 5];not only primes/proper divisors or all integers<=n

Conditions: none

Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.

Step identifiers and dependencies

Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S047

S049 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor := mul(5, 1)

After

Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor = 5

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b := sum(5, 7)

After

Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b = 12

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor := mul(5, 5)

After

Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor = 25

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b := sum(25, 7)

After

Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b = 32

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum := sum(12, 32)

After

Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum = 44

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) := mul(0, 44)

After

Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) = 0

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor5:actual positive input for mu=5

After

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Conditions: none

Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.

Step identifiers and dependencies

Rule: mobius_complete_UPF_input_role · Uses: G001, S054

S056 · implies

Before

ORIGINAL positive integer absolute magnitude=5>1

After

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

Conditions: none

Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.

Step identifiers and dependencies

Rule: upf_factorization_remaining_product_invariant · Uses: G001, S055

S057 · implies

Before

next divisor=2;previous remaining=5;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=5;remaining modulo2=1;extracted multiset=[]

Conditions: none

Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.

Step identifiers and dependencies

Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S056

S058 · calculation

Before

upf canonical complete product factor0 := mul(1, 5)

After

upf canonical complete product factor0 = 5

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · implies

Before

exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=5

After

complete increasing prime multiset=[5];full product=5;a composite terminal remainder would have an already tested smaller divisor

Conditions: none

If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.

Step identifiers and dependencies

Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S058

S060 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor5:input=5;COMPLETE prime multiset=[5];distinct primes=[5]

After

mu=-1;empty product case=No;repeated prime=No

Conditions: none

The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.

Step identifiers and dependencies

Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S059

S061 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 := div(20, 5)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 = 4

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · implies

Before

Mobius inversion ORIGINAL inverse F(n/5):actual positive input=4;check complete integer range1..4

After

ALL positive divisors INCREASING=[1, 2, 4];not only primes/proper divisors or all integers<=n

Conditions: none

Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.

Step identifiers and dependencies

Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S061

S063 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor := mul(5, 1)

After

Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor = 5

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b := sum(5, 7)

After

Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b = 12

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor := mul(5, 2)

After

Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor = 10

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b := sum(10, 7)

After

Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b = 17

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor := mul(5, 4)

After

Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor = 20

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b := sum(20, 7)

After

Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b = 27

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum := sum(12, 17, 27)

After

Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum = 56

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · calculation

Before

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) := mul(-1, 56)

After

Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) = -56

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor10:actual positive input for mu=10

After

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Conditions: none

Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.

Step identifiers and dependencies

Rule: mobius_complete_UPF_input_role · Uses: G001, S070

S072 · implies

Before

ORIGINAL positive integer absolute magnitude=10>1

After

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

Conditions: none

Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.

Step identifiers and dependencies

Rule: upf_factorization_remaining_product_invariant · Uses: G001, S071

S073 · calculation

Before

upf extract divisor2 occurrence0 := div(10, 2)

After

upf extract divisor2 occurrence0 = 5

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · implies

Before

next divisor=2;previous remaining=10;all smaller divisors exhausted

After

extracted multiplicity=1;next remaining=5;remaining modulo2=1;extracted multiset=[2]

Conditions: none

Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.

Step identifiers and dependencies

Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S073

S075 · calculation

Before

upf canonical complete product factor0 := mul(1, 2)

After

upf canonical complete product factor0 = 2

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · calculation

Before

upf canonical complete product factor1 := mul(2, 5)

After

upf canonical complete product factor1 = 10

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · implies

Before

exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=10

After

complete increasing prime multiset=[2, 5];full product=10;a composite terminal remainder would have an already tested smaller divisor

Conditions: none

If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.

Step identifiers and dependencies

Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S076

S078 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor10:input=10;COMPLETE prime multiset=[2, 5];distinct primes=[2, 5]

After

mu=1;empty product case=No;repeated prime=No

Conditions: none

The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.

Step identifiers and dependencies

Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S077

S079 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 := div(20, 10)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 = 2

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · implies

Before

Mobius inversion ORIGINAL inverse F(n/10):actual positive input=2;check complete integer range1..2

After

ALL positive divisors INCREASING=[1, 2];not only primes/proper divisors or all integers<=n

Conditions: none

Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.

Step identifiers and dependencies

Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S079

S081 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor := mul(5, 1)

After

Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor = 5

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b := sum(5, 7)

After

Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b = 12

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S081

S083 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor := mul(5, 2)

After

Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor = 10

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b := sum(10, 7)

After

Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b = 17

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S083

S085 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum := sum(12, 17)

After

Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum = 29

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S084

S086 · calculation

Before

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) := mul(1, 29)

After

Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) = 29

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S085

S087 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor20:actual positive input for mu=20

After

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Conditions: none

Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.

Step identifiers and dependencies

Rule: mobius_complete_UPF_input_role · Uses: G001, S086

S088 · implies

Before

ORIGINAL positive integer absolute magnitude=20>1

After

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

Conditions: none

Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.

Step identifiers and dependencies

Rule: upf_factorization_remaining_product_invariant · Uses: G001, S087

S089 · calculation

Before

upf extract divisor2 occurrence0 := div(20, 2)

After

upf extract divisor2 occurrence0 = 10

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · calculation

Before

upf extract divisor2 occurrence1 := div(10, 2)

After

upf extract divisor2 occurrence1 = 5

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S089

S091 · implies

Before

next divisor=2;previous remaining=20;all smaller divisors exhausted

After

extracted multiplicity=2;next remaining=5;remaining modulo2=1;extracted multiset=[2, 2]

Conditions: none

Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted.

Step identifiers and dependencies

Rule: upf_complete_trial_division_remaining_invariant · Uses: G001, S090

S092 · calculation

Before

upf canonical complete product factor0 := mul(1, 2)

After

upf canonical complete product factor0 = 2

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S091

S093 · calculation

Before

upf canonical complete product factor1 := mul(2, 2)

After

upf canonical complete product factor1 = 4

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S092

S094 · calculation

Before

upf canonical complete product factor2 := mul(4, 5)

After

upf canonical complete product factor2 = 20

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S093

S095 · implies

Before

exhausted trial divisors below3;terminal remaining=5;d*d>remaining;ORIGINAL n=20

After

complete increasing prime multiset=[2, 2, 5];full product=20;a composite terminal remainder would have an already tested smaller divisor

Conditions: none

If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.

Step identifiers and dependencies

Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S094

S096 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor20:input=20;COMPLETE prime multiset=[2, 2, 5];distinct primes=[2, 5]

After

mu=0;empty product case=No;repeated prime=Yes

Conditions: none

The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.

Step identifiers and dependencies

Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S095

S097 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 := div(20, 20)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 = 1

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S096

S098 · implies

Before

Mobius inversion ORIGINAL inverse F(n/20):actual positive input=1;check complete integer range1..1

After

ALL positive divisors INCREASING=[1];not only primes/proper divisors or all integers<=n

Conditions: none

Check divisibility over the complete range,retaining1 and n.The relation d|n is neither d<=n nor a prime/proper-divisor restriction.

Step identifiers and dependencies

Rule: mobinv_ALL_positive_divisors_complete_indexing · Uses: G001, S097

S099 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor := mul(5, 1)

After

Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor = 5

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b := sum(5, 7)

After

Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b = 12

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S099

S101 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum := sum(12)

After

Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum = 12

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S100

S102 · calculation

Before

Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) := mul(0, 12)

After

Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) = 0

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S101

S103 · calculation

Before

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(252, -118, 0, -56, 29, 0)

After

Mobius inversion ORIGINAL inverse complete inverse-term sum = 107

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S102

S104 · calculation

Before

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(5, 20)

After

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 100

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S103

S105 · calculation

Before

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b := sum(100, 7)

After

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 107

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S104

S106 · implies

Before

Mobius inversion ORIGINAL inverse:actual n=20;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 20, 252, 252], [2, -1, 10, 118, -118], [4, 0, 5, 44, 0], [5, -1, 4, 56, -56], [10, 1, 2, 29, 29], [20, 0, 1, 12, 0]]

After

complete reconstructed=107;direct f(n)=107;sum_d mu(d)sum_e|n/d f(e)=sum_e|n f(e)sum_d|n/e mu(d)=f(n)

Conditions: none

Swap the finite sums over d*e|n.The inner Mobius sum is1 for n/e=1 and0 otherwise by pairing distinct-prime subsets,leaving only e=n.For n1,the d1 term and mu1=1 remain;do not mix indices from the alternate equivalent convention.

Step identifiers and dependencies

Rule: mobinv_general_finite_double_sum_and_Mobius_cancellation · Uses: G001, S105

Reverse checks

  • S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S105 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S104 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
  • S103 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
  • S102 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(20) times SAME F(n/20) with exact rational arithmetic, not rounded decimals.
  • S101 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) full forward divisor sum with exact rational arithmetic, not rounded decimals.
  • S100 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S099 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/20) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S097 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor20 with exact rational arithmetic, not rounded decimals.
  • S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S094 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals.
  • S093 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
  • S092 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
  • S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S090 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
  • S089 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
  • S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S086 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(10) times SAME F(n/10) with exact rational arithmetic, not rounded decimals.
  • S085 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) full forward divisor sum with exact rational arithmetic, not rounded decimals.
  • S084 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S083 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S082 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S081 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/10) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S079 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor10 with exact rational arithmetic, not rounded decimals.
  • S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S076 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
  • S075 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
  • S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S073 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
  • S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S070 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(5) times SAME F(n/5) with exact rational arithmetic, not rounded decimals.
  • S069 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) full forward divisor sum with exact rational arithmetic, not rounded decimals.
  • S068 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S067 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S066 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S065 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S064 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S063 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/5) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S061 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor5 with exact rational arithmetic, not rounded decimals.
  • S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S058 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
  • S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S054 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(4) times SAME F(n/4) with exact rational arithmetic, not rounded decimals.
  • S053 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) full forward divisor sum with exact rational arithmetic, not rounded decimals.
  • S052 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S051 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S050 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S049 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/4) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor4 with exact rational arithmetic, not rounded decimals.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals.
  • S043 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Recompute upf extract divisor2 occurrence1 with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute upf extract divisor2 occurrence0 with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(2) times SAME F(n/2) with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) full forward divisor sum with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/2) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor2 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(20) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(20) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(10) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(10) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(5) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(5) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(20, [[1, 1, 20, 252, 252], [2, -1, 10, 118, -118], [4, 0, 5, 44, 0], [5, -1, 4, 56, -56], [10, 1, 2, 29, 29], [20, 0, 1, 12, 0]], 107, 107, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 34be98f6fd4b8088085494369cb9f4182e392a5c7e878e695c81d1dd1951ea00
  • candidate record SHA-256: 2b4d1343a45e7818abc64cdfd3943d6a5878dca76b96832584dda6b6e47f9791
  • semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
  • question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON