cp's OEIS Frontend

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.

%I A316274 #15 Nov 15 2018 00:50:19
%S A316274 1,-4,-16,-9,-48,-25,-54,-128,36,-49,-320,144,-243,100,216,-121,-250,
%T A316274 -768,432,-169,196,400,864,225,-972,-1792,1152,-289,972,-686,-361,784,
%U A316274 1200,2592,441,484,1000
%N 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.
%C A316274 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".
%H A316274 Mats Granvik, <a href="https://mathoverflow.net/q/299533/25104">Does the Dirichlet Inverse of the Euler totient function characterize asymptotic densities of prime gaps?</a>.
%H A316274 Terence Tao, <a href="https://terrytao.wordpress.com/2017/07/06/correlations-of-the-von-mangoldt-and-higher-divisor-functions-i-long-shift-ranges/">Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges</a>.
%F A316274 a(n) = A298825(A001694(n)). - _Mats Granvik_, Oct 08 2018
%t A316274 Clear[nn, h, a, n, d, b, m];
%t A316274 nn = 500;
%t A316274 h = 1;
%t A316274 a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]];
%t A316274 TableForm[Transpose[Table[{n, a[n]}, {n, 1, nn}]]];
%t A316274 b = DeleteCases[
%t A316274   Table[Sum[
%t A316274     Sum[If[Mod[n, k] == 0, a[GCD[n/k, m]]*a[GCD[k, m + h]], 0], {k, 1,
%t A316274        n}], {m, 1, n}], {n, 1, nn}], 0]
%Y A316274 Cf. A191898.
%K A316274 sign
%O A316274 1,2
%A A316274 _Mats Granvik_, Jun 28 2018