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

1,1

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

Formula

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

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1<d<n} A347236(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)).

Prog

(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(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A003961(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));

Crossrefs

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

Sequence in context: A354876 A323408 A347099 * A347097 A339274 A335156

Adjacent sequences: A347236 A347237 A347238 * A347240 A347241 A347242

Keyword

nonn

Author

Antti Karttunen, Aug 24 2021

Status

approved