bibkey: granvik2011a192763
authors: Mats Granvik; Olivier Gérard; Michael Somos
year: 2011
title: "OEIS A192763 and the Granvik Mertens array"
doi: null
url: https://oeis.org/A192763
claim: "The main diagonal is the Mobius function times the natural numbers A055615 (conjecture). For k>1 the first row is the Mertens function + 2 = A002321 + 2 (conjecture)."
strata_touched:
  - D5/S3/Arith/GranvikMertensArray
license: citation-only
triage: anchor

The Granvik Mertens array

Mats Granvik’s OEIS A192763 COMMENTS field states:

The main diagonal is the Mobius function times the natural numbers A055615 (conjecture). For k>1 the first row is the Mertens function + 2 = A002321 + 2 (conjecture).

The entry’s first Mathematica program, attributed to Mats Granvik and Olivier Gérard, sets the corner to one and the next two border cells to two. Michael Somos’s Mathematica and PARI programs give the positive Euclidean reduction $T(n,k)=T(k,((n-1)\mathrm{mod}k)+1)$ for $n>k>1$, the zero row-sum diagonal, symmetry, and the border update used in the formal definition. All division is interpreted in $\mathbb{Q}$.

Verified locator

Declared source: https://oeis.org/A192763

Pinned official export: https://github.com/oeis/oeisdata/blob/0b811e71e00a3e9b1374c94971f83d8e4c8af516/seq/A192/A192763.seq

Companion question by Mats Granvik, “Is this a recurrence for the Mertens function plus 2?”: https://math.stackexchange.com/questions/50719/

The pinned export contains the conjecture in COMMENTS, Somos’s recurrence in his Mathematica and PARI programs, and the reverse-window recurrence in the first Mathematica program. The Math.SE question prints a corner value of three; the OEIS programs use one. The settlement uses the OEIS normalization. The question’s displayed formulas contain index slips, so the executable OEIS programs fix the recurrence interpreted here.

Settlement scope

GranvikMertensArray.result states the diagonal and first-row identities for all positive diagonal indices and all first-row indices at least two. Its proof includes equality with the literal first Mathematica recurrence, by well-founded uniqueness. The extension at a zero index is zero and is outside the published positive-index array.

For coprime interior indices, signed Bézout descent identifies the terminal Euclidean quotient with the quotient determined by the smaller inverse residue. Inversion permutes reduced residues. Möbius convolution then evaluates their border sum, and grouping the other cells by their gcd gives the full row sum. This arithmetic identity forces the border update.

The source remains labelled conjectural in the pinned export. The Stack Exchange API reading on 2026-10-23 reports zero answers and is_answered: false. These bounded readings do not establish literature completeness or publication priority, both ASSUMED-UNVERIFIED. No asymptotic bound for the Mertens function is asserted.