A347229 Sum of A344695 [gcd(sigma(n), psi(n))] and its Dirichlet inverse.

2, 0, 0, 9, 0, 24, 0, -21, 16, 36, 0, -28, 0, 48, 48, 73, 0, -42, 0, -42, 64, 72, 0, 108, 36, 84, -56, -56, 0, 0, 0, -213, 96, 108, 96, 121, 0, 120, 112, 162, 0, 0, 0, -84, -84, 144, 0, -284, 64, -102, 144, -98, 0, 192, 144, 216, 160, 180, 0, 216, 0, 192, -112, 649, 168, 0, 0, -126, 192, 0, 0, -357, 0, 228, -136

Offset: 1

Author

Antti Karttunen, Aug 25 2021

Keywords

• sign

Comments

It seems that A030059 gives the positions of all zeros.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..12000

Crossrefs

Cf. A000203, A001615, A030059, A030229, A344695, A347228, A347231.

Programs

PARI

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344695(n) = gcd(sigma(n), A001615(n));
v347228 = DirInverseCorrect(vector(up_to,n,A344695(n)));
A347228(n) = v347228[n];
A347229(n) = (A344695(n)+A347228(n));

Formula

a(n) = A344695(n) + A347228(n).

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A344695(d) * A347228(n/d).

For all n >= 1, a(A030059(n)) = 0 and a(A030229(n)) = 2*A344695(A030229(n)). [Even though A344695 is not multiplicative, this holds because on squarefree n it is equal to psi(n) and sigma(n) that are multiplicative functions]