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Question 22124: The Forward Transform Sums over All Positive Divisors

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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R0043-22124 · q-20261014-advance-20261110-r06-mobius-inversion-finite-divisors-0024

Original question and givens

Use positive integer n=24, exact integer arithmetic function f(k)=a*k+b with a=4, b=11. 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, divisor_value_rows, F_n, certificate).

  • G001 — Use positive integer n=24, exact integer arithmetic function f(k)=a*k+b with a=4, b=11. 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, divisor_value_rows, F_n, certificate).

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-forward-divisor-transform;original snapshot and mathematics retained

After

explicit ordered source output fields=[n, divisor_value_rows, F_n, certificate]

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 forward:actual positive input=24;check complete integer range1..24

After

ALL positive divisors INCREASING=[1, 2, 3, 4, 6, 8, 12, 24];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 · calculation

Before

Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor := mul(4, 1)

After

Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor = 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, S002

S004 · calculation

Before

Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b := sum(4, 11)

After

Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b = 15

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, S003

S005 · calculation

Before

Mobius inversion ORIGINAL forward f(2) ORIGINAL a times divisor := mul(4, 2)

After

Mobius inversion ORIGINAL forward f(2) ORIGINAL a times divisor = 8

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 · calculation

Before

Mobius inversion ORIGINAL forward f(2) plus ORIGINAL b := sum(8, 11)

After

Mobius inversion ORIGINAL forward f(2) plus ORIGINAL b = 19

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, S005

S007 · calculation

Before

Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor := mul(4, 3)

After

Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor = 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, S006

S008 · calculation

Before

Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b := sum(12, 11)

After

Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b = 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, S007

S009 · calculation

Before

Mobius inversion ORIGINAL forward f(4) ORIGINAL a times divisor := mul(4, 4)

After

Mobius inversion ORIGINAL forward f(4) ORIGINAL a times divisor = 16

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 forward f(4) plus ORIGINAL b := sum(16, 11)

After

Mobius inversion ORIGINAL forward 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, S009

S011 · calculation

Before

Mobius inversion ORIGINAL forward f(6) ORIGINAL a times divisor := mul(4, 6)

After

Mobius inversion ORIGINAL forward f(6) ORIGINAL a times divisor = 24

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 forward f(6) plus ORIGINAL b := sum(24, 11)

After

Mobius inversion ORIGINAL forward f(6) plus ORIGINAL b = 35

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 forward f(8) ORIGINAL a times divisor := mul(4, 8)

After

Mobius inversion ORIGINAL forward f(8) ORIGINAL a times divisor = 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, S012

S014 · calculation

Before

Mobius inversion ORIGINAL forward f(8) plus ORIGINAL b := sum(32, 11)

After

Mobius inversion ORIGINAL forward f(8) plus ORIGINAL b = 43

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 forward f(12) ORIGINAL a times divisor := mul(4, 12)

After

Mobius inversion ORIGINAL forward f(12) ORIGINAL a times divisor = 48

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 forward f(12) plus ORIGINAL b := sum(48, 11)

After

Mobius inversion ORIGINAL forward f(12) plus ORIGINAL b = 59

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 forward f(24) ORIGINAL a times divisor := mul(4, 24)

After

Mobius inversion ORIGINAL forward f(24) ORIGINAL a times divisor = 96

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 forward f(24) plus ORIGINAL b := sum(96, 11)

After

Mobius inversion ORIGINAL forward f(24) 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 forward full forward divisor sum := sum(15, 19, 23, 27, 35, 43, 59, 107)

After

Mobius inversion ORIGINAL forward full forward divisor sum = 328

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

Reverse checks

  • S019 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward full forward divisor sum with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(24) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(24) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(12) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(12) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(8) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(8) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(6) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(6) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(4) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(4) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(3) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(3) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(2) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(2) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(1) plus ORIGINAL b with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Mobius inversion ORIGINAL forward f(1) ORIGINAL a times divisor with exact rational arithmetic, not rounded decimals.
  • 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

(24, [[1, 15], [2, 19], [3, 23], [4, 27], [6, 35], [8, 43], [12, 59], [24, 107]], 328, sum_over_all_positive_divisors)

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: fecfa2d905b81bf58634043ba32444f4c648e1aa94220c092f8c114a89e5267c
  • candidate record SHA-256: 94736cc1c2269cb11600ba17be338f6bd61860ff034b2f5b70c245b3a521ca01
  • semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
  • question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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