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Question 22173: 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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R0043-22173 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0048

Original question and givens

Use positive integer n=23, exact integer arithmetic function f(k)=a*k+b with a=3, b=10. 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=23, exact integer arithmetic function f(k)=a*k+b with a=3, b=10. 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=23;check complete integer range1..23

After

ALL positive divisors INCREASING=[1, 23];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(23, 1)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor1 = 23

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=23;check complete integer range1..23

After

ALL positive divisors INCREASING=[1, 23];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(3, 1)

After

Mobius inversion ORIGINAL inverse F(n/1) f(1) ORIGINAL a times divisor = 3

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(3, 10)

After

Mobius inversion ORIGINAL inverse F(n/1) f(1) plus ORIGINAL b = 13

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(23) ORIGINAL a times divisor := mul(3, 23)

After

Mobius inversion ORIGINAL inverse F(n/1) f(23) ORIGINAL a times divisor = 69

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(23) plus ORIGINAL b := sum(69, 10)

After

Mobius inversion ORIGINAL inverse F(n/1) f(23) plus ORIGINAL b = 79

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) full forward divisor sum := sum(13, 79)

After

Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum = 92

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 mu(1) times SAME F(n/1) := mul(1, 92)

After

Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) = 92

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 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor23:actual positive input for mu=23

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, S012

S014 · implies

Before

ORIGINAL positive integer absolute magnitude=23>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, S013

S015 · implies

Before

next divisor=2;previous remaining=23;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=23;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, S014

S016 · implies

Before

next divisor=3;previous remaining=23;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=23;remaining modulo3=2;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, S015

S017 · implies

Before

next divisor=4;previous remaining=23;all smaller divisors exhausted

After

extracted multiplicity=0;next remaining=23;remaining modulo4=3;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, S016

S018 · calculation

Before

upf canonical complete product factor0 := mul(1, 23)

After

upf canonical complete product factor0 = 23

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, S017

S019 · implies

Before

exhausted trial divisors below5;terminal remaining=23;d*d>remaining;ORIGINAL n=23

After

complete increasing prime multiset=[23];full product=23;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, S018

S020 · implies

Before

Mobius inversion ORIGINAL inverse Mobius at ORIGINAL divisor23:input=23;COMPLETE prime multiset=[23];distinct primes=[23]

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, S019

S021 · calculation

Before

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23 := div(23, 23)

After

Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23 = 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, S020

S022 · implies

Before

Mobius inversion ORIGINAL inverse F(n/23):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, S021

S023 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/23) f(1) ORIGINAL a times divisor := mul(3, 1)

After

Mobius inversion ORIGINAL inverse F(n/23) f(1) ORIGINAL a times divisor = 3

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, S022

S024 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b := sum(3, 10)

After

Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b = 13

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, S023

S025 · calculation

Before

Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum := sum(13)

After

Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum = 13

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, S024

S026 · calculation

Before

Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23) := mul(-1, 13)

After

Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23) = -13

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 · calculation

Before

Mobius inversion ORIGINAL inverse complete inverse-term sum := sum(92, -13)

After

Mobius inversion ORIGINAL inverse complete inverse-term sum = 79

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, S026

S028 · calculation

Before

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n := mul(3, 23)

After

Mobius inversion ORIGINAL inverse direct ORIGINAL a times n = 69

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 direct plus ORIGINAL b := sum(69, 10)

After

Mobius inversion ORIGINAL inverse direct plus ORIGINAL b = 79

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 · implies

Before

Mobius inversion ORIGINAL inverse:actual n=23;ALL [d,mu(d),n/d,F(n/d),mu(d)*F(n/d)] rows=[[1, 1, 23, 92, 92], [23, -1, 1, 13, -13]]

After

complete reconstructed=79;direct f(n)=79;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, S029

Reverse checks

  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S028 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse direct ORIGINAL a times n with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse complete inverse-term sum with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(23) times SAME F(n/23) with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/23) full forward divisor sum with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/23) f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/23) f(1) ORIGINAL a times divisor 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 — Recompute Mobius inversion ORIGINAL inverse n divided by ORIGINAL divisor23 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse mu(1) times SAME F(n/1) with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) full forward divisor sum with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(23) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL inverse F(n/1) f(23) 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

(23, [[1, 1, 23, 92, 92], [23, -1, 1, 13, -13]], 79, 79, 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: 36a10cec56c0331f023b8bed319aa15c5eac68aefa77809378b3384d296bfb88
  • candidate record SHA-256: 5843563c7b59ec768bf3fffeb7d2a0b9a70344e2ba4c78abc318c42e863106dc
  • semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
  • question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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