A316274 Nonzero terms in row sums of the lower triangular part of a square matrix formed by Dirichlet convolution of adjacent columns in the square matrix A191898.

1, -4, -16, -9, -48, -25, -54, -128, 36, -49, -320, 144, -243, 100, 216, -121, -250, -768, 432, -169, 196, 400, 864, 225, -972, -1792, 1152, -289, 972, -686, -361, 784, 1200, 2592, 441, 484, 1000

Offset: 1

Author

Mats Granvik, Jun 28 2018

Keywords

• sign

Comments

The motivation for this sequence is expression 1 in Terence Tao's blog post "Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges".

Links

• Mats Granvik, Does the Dirichlet Inverse of the Euler totient function characterize asymptotic densities of prime gaps?.
• Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges.

Crossrefs

Cf. A191898.

Programs

Mathematica

Clear[nn, h, a, n, d, b, m];
nn = 500;
h = 1;
a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]];
TableForm[Transpose[Table[{n, a[n]}, {n, 1, nn}]]];
b = DeleteCases[
  Table[Sum[
    Sum[If[Mod[n, k] == 0, a[GCD[n/k, m]]*a[GCD[k, m + h]], 0], {k, 1,
       n}], {m, 1, n}], {n, 1, nn}], 0]

Formula

a(n) = A298825(A001694(n)). - Mats Granvik, Oct 08 2018