Metadata
bibkey: oeis2026a400361
authors: Ctibor O. Zizka; OEIS Foundation Inc.
year: 2026
title: OEIS A400361 revision 17, the second-order Pillai minus floor-gcd conjecture
doi: null
url: https://oeis.org/A400361
claim: The unrestricted second-order cumulative formula with the literal digamma, totient, and zeta-derivative constant.
strata_touched:
- D5/S3/ArithSums/A400361SecondTerm
license: citation-only
triage: anchor
A400361 and the exact second-order constant
Primary statement
The authoritative source is OEIS A400361 revision 17,
with machine-readable text at https://oeis.org/search?q=id:A400361&fmt=text.
Its NAME is a(n) = A018804(n) - A384628(n). The offset is one.
A018804 is Pillai’s sum of gcd(n,k), and
A384628 is the sum of gcd(n,floor(n/k)), both for
1 <= k <= n. Subtraction is signed before the real embedding.
The revision-17 COMMENT states:
Conjecture: Sum_{k=1..n} a(k) = (3/Pi^2)n^2log(n) + Cn^2 + o(n^2), with C = (3/Pi^2)(-1/2 + 2gamma - 6zeta’(2)/Pi^2) - Pi^2/12 - (1/2)Sum_{d>=1} phi(d)(digamma(1+1/d) + gamma)/d^2 = -1.2036….
The entry credits Ctibor O. Zizka with the sequence. The conjecture line has no individual attribution; no conjecture-author credit is inferred from other entries. gamma is the Euler-Mascheroni constant, phi is Euler’s totient, and digamma is the logarithmic derivative of Gamma. The displayed decimal is source context and is not a numerical certification here.
Verified locator
The complete primary entry is OEIS A400361.
The revision-17 full-content export
contains the NAME, COMMENT conjecture, positive offset and source attribution.
Its 1,750 UTF-8 bytes have SHA-256
52a8b75baa111c1d3c2e01a545885b65f5234e59b5949572edb10e6cf49ed1e7.
The COMMENT beginning Conjecture: Sum_{k=1..n} a(k) is the assertion
quoted above; the related positive-index definitions are A018804 and A384628.
Classical part and attribution
Laszlo Toth, A Survey of Gcd-Sum Functions, Journal of Integer Sequences 13 (2010), Article 10.8.1, gives the classical Pillai mean with secondary coefficient
Original text: https://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.html and https://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.tex. This Pillai asymptotic and the Mobius/hyperbola methods are prior mathematics, not a separate problem resolution or priority claim. The native proof uses the existing Pillai divisor formula, Mathlib’s harmonic-number limit and zeta-Mobius Dirichlet identity, then proves the required transfer estimates. No external proof code is reproduced.
Actual literature scope
The bounded qualification inspected the actual relevant scopes of Toth, Bordelles, Abel-Awan-Kushnirevych, and Chaubey-Goel. No equivalent whole second-order A400361 assertion was established there. A conflicting introductory Chaubey-Goel quotation is excluded from the mathematical basis. The Bu-Xu full text was unavailable (HTTP 403); its abstract does not settle the overlap question. That full-text overlap remains unknown. These source nonhits do not establish global absence, exhaustive clearance, priority, or an independent discovery claim.
Mathematical correspondence
The real restriction of Riemann zeta has derivative at two equal to the real part of its complex derivative. At the positive arguments 1+1/d, the derivative quotient of the real Gamma function equals complex digamma. The proof includes both correspondences. Every positive d, including one, occurs once as j+1. The coefficient series is absolutely convergent, with nonnegative summand bounded by pi^2/(6*d^2). The final remainder is little-o along all natural indices at infinity, hence along all positive integers. No stronger effective error term or numerical value of C is asserted.
License
OEIS content is licensed under CC BY-SA 4.0; see https://oeis.org/wiki/The_OEIS_End-User_License_Agreement. The source quotation retains its OEIS attribution. Toth’s paper is cited for its theorem and methods. All Lean proof code in this delivery is native code using the pinned project and Mathlib declarations.