A347239 Sum of A347236 and its Dirichlet inverse.

2, 0, 0, 1, 0, 4, 0, 13, 4, 4, 0, 26, 0, 8, 8, 55, 0, 34, 0, 26, 16, 4, 0, 26, 4, 8, 68, 52, 0, 0, 0, 133, 8, 4, 16, 223, 0, 8, 16, 26, 0, 0, 0, 26, 68, 12, 0, 110, 16, 74, 8, 52, 0, 68, 8, 52, 16, 4, 0, 4, 0, 12, 136, 463, 16, 0, 0, 26, 24, 0, 0, 247, 0, 8, 148, 52, 16, 0, 0, 110, 421, 4, 0, 8, 8, 8, 8, 26, 0, 8, 32

Offset: 1

Author

Antti Karttunen, Aug 24 2021

Keywords

• nonn

Comments

It seems that A030059 gives the positions of all zeros.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000290, A001223, A001248, A003961, A061019, A030059, A030229, A347236, A347238.

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(dA003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));
v347238 = DirInverseCorrect(vector(up_to,n,A347236(n)));
A347238(n) = v347238[n];
A347239(n) = (A347236(n)+A347238(n));

Formula

a(n) = A347236(n) + A347238(n).

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A347236(d) * A347238(n/d).

For all n >= 1, a(A030059(n)) = 0 and a(A030229(n)) = 2*A347236(A030229(n)).

For all n >= 1, a(A001248(n)) = A000290(A001223(n)).