A342419 Sum of A342002 and its Dirichlet inverse.

2, 0, 0, 1, 0, 10, 0, 3, 25, 14, 0, 7, 0, 14, 70, 15, 0, -3, 0, 33, 70, 82, 0, 45, 49, 18, 185, 29, 0, 8, 0, 35, 410, 94, 98, 28, 0, 22, 90, 97, 0, 0, 0, 79, 279, 106, 0, -27, 49, 61, 470, 43, 0, -315, 574, 111, 110, 118, 0, -199, 0, 18, 339, 43, 126, -876, 0, 81, 530, 152, 0, 6, 0, 118, -87, 153, 574, 412, 0, 267

Offset: 1

Author

Antti Karttunen, Mar 13 2021

Keywords

• sign

Comments

Zeros occurring on composite n are rare: 42 and 4718 are the first two such positions.

It seems that many nonzero squares occur on square n.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..11550

Crossrefs

Cf. A342002, A342417.

Programs

PARI

up_to = 11550;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
v342417 = DirInverseCorrect(vector(up_to,n,A342002(n)));
A342417(n) = v342417[n];
A342419(n) = (A342002(n)+A342417(n));

Formula

a(n) = A342002(n) + A342417(n).