A347228 Dirichlet inverse of A344695, gcd(sigma(n), psi(n)).

1, -3, -4, 8, -6, 12, -8, -24, 15, 18, -12, -32, -14, 24, 24, 72, -18, -45, -20, -48, 32, 36, -24, 96, 35, 42, -60, -64, -30, -72, -32, -216, 48, 54, 48, 120, -38, 60, 56, 144, -42, -96, -44, -96, -90, 72, -48, -288, 63, -105, 72, -112, -54, 180, 72, 192, 80, 90, -60, 192, -62, 96, -120, 648, 84, -144, -68, -144, 96

Offset: 1

Author

Antti Karttunen, Aug 25 2021

Keywords

• sign

Comments

This is not multiplicative because A344695 is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = -484 <> -480 = 8 * -60 = a(4) * a(27).

Conjecture: No zeros occur as terms. Checked up to n = 2^21.

Links

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

Crossrefs

Cf. A000203, A001615, A344695, A347229, A347230.

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];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A344695(n/d) * a(d).

a(n) = A347229(n) - A344695(n).