A347094 Sum of A038040 (convolution of sigma with Euler phi) and its Dirichlet inverse.

2, 0, 0, 16, 0, 48, 0, 32, 36, 80, 0, 48, 0, 112, 120, 80, 0, 72, 0, 80, 168, 176, 0, 192, 100, 208, 108, 112, 0, 0, 0, 192, 264, 272, 280, 360, 0, 304, 312, 320, 0, 0, 0, 176, 180, 368, 0, 480, 196, 200, 408, 208, 0, 432, 440, 448, 456, 464, 0, 960, 0, 496, 252, 448, 520, 0, 0, 272, 552, 0, 0, 864, 0, 592, 300, 304, 616

Offset: 1

Author

Antti Karttunen, Aug 18 2021

Keywords

• nonn

Comments

No negative terms in range 1 .. 2^20.

Apparently, A030059 gives the positions of all zeros.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000005, A030059, A030229, A038040, A328722, A347093.

Programs

PARI

up_to = 16384;
A038040(n) = (n*numdiv(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA038040(n)));
A328722(n) = v328722[n];
A347094(n) = (A038040(n)+A328722(n));

Formula

a(n) = A038040(n) + A328722(n).

For n > 1, a(n) = -Sum_{d|n, 1A038040(d) * A328722(n/d).

For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A038040(A030229(n)).