slug: oeis-a344598-alternating-gcd-sum-pillai bibkey: bala2024a344598 doi: null url: https://oeis.org/A344598 triage: theorem motivation_gids:

• D5/S3/Factorization/AlternatingGcdSumPillai

Bala’s alternating gcd-sum formula for A344598

Problem

OEIS A344598 NAME (Seiichi Manyama, May 24 2021):

a(n) = Sum_{k=1..n} phi(k) * (floor(n/k)^2 - floor((n-1)/k)^2).

FORMULA (Peter Bala, Jan 01 2024):

Conjecture: a(n) = Sum_{k = 1..2n} (-1)^k * gcd(k, 4n). Cf. A344372. - Peter Bala, Jan 01 2024

Both quotations are copied verbatim from the claim in Library/Arith/bala2024a344598.md. The target is the equality for every positive n.

范围与诚实边界: This is a modest identity (tier 1). The two divisor-sum formulas a(n) = Sum_{d|n} phi(d)*(2*n/d - 1) and a(n) = Sum_{d|n} (2*d*tau(d) - sigma(d))*mu(n/d) were already proved elsewhere by Daniel Weber, as credited in Mikhail Kurkov’s Mar 31 2026 comment. They are not the target. The target is the alternating-gcd formula, which the search seat’s 2026-09-08 reading recorded as still marked Conjecture. Natural division and truncated subtraction define a; the alternating sum and theorem subtraction are in the integers.

Motivation

This is a first-tier recent OEIS conjecture. KPI = open problems resolved, not module counts or a finite prefix checked. The module proves the assertion for every positive natural-number index.

Gap

The supplied search reports no proof found: the search seat read the OEIS entry and revision history on 2026-09-08 and searched the identifier on arXiv, MathOverflow, and GitHub. This Stage-B seat has no network and did not repeat those searches. No-proof-found is restricted to that reported search scope.

Route

The totient bridge a_eq_two_pillai_sub proves a(n) = 2*pillai(n) - n: floor-square increments select the divisors of n, gcd fibres give the totient-weighted quotient sum, and Mathlib Nat.sum_totient supplies n.

Two gcd bookkeeping identities finish the proof. pillai_four_mul proves pillai(4n) + 4*pillai(n) = 4*pillai(2n) by an even/odd range decomposition and equality of shifted odd-index gcd blocks. altGcdSum_eq proves 2*altGcdSum(n) + 2n = 4*pillai(2n) - pillai(4n) by a reflected-index construction with endpoint accounting. Combining these identities gives bala_conjecture. Imports are Mathlib-only; generality is G.

Falsifier

A positive natural-number index n for which the defining totient sum differs from the alternating gcd sum would contradict the assertion. The orchestrator’s exact numerical check is supporting evidence only, not the universal proof.

Evidence

• Lean module: D5/S3/Factorization/AlternatingGcdSumPillai.lean.
• Main public theorem: bala_conjecture.
• Companions: a_eq_two_pillai_sub, pillai_four_mul, altGcdSum_eq.
• Axioms: std3 (propext, Classical.choice, Quot.sound) for each public theorem, as recorded by the implementation seat’s #print axioms receipt.
• The main theorem quantifies over all n : ℕ with 1 ≤ n; no finite search bound is a hypothesis.

Triage

theorem. The formal proof closes Bala’s universal alternating-gcd assertion.

ASSUMED-UNVERIFIED

The OEIS quotes and attributions were supplied by the orchestrator and copied from the Library note. OEIS revision history was read by the search seat on 2026-09-08, not by this seat. Literature scope is the OEIS entry and history plus identifier searches on arXiv, MathOverflow, and GitHub, as reported above; it is not an exhaustive literature search or a claim of first-publication priority. This seat did not independently verify the online sources, the orchestrator’s numerical check, or the implementation seat’s axiom receipt. Source-to-Lean fidelity and publication priority are not kernel-checked facts.