A378607 Dirichlet convolution of sigma and the Dirichlet inverse of A003961 (A346234).

1, 0, -1, -2, -1, 0, -3, -6, -7, 0, -1, 2, -3, 0, 1, -14, -1, 0, -3, 2, 3, 0, -5, 6, -11, 0, -25, 6, -1, 0, -5, -30, 1, 0, 3, 14, -3, 0, 3, 6, -1, 0, -3, 2, 7, 0, -5, 14, -31, 0, 1, 6, -5, 0, 1, 18, 3, 0, -1, -2, -5, 0, 21, -62, 3, 0, -3, 2, 5, 0, -1, 42, -5, 0, 11, 6, 3, 0, -3, 14, -79, 0, -5, -6, 1, 0, 1, 6, -7, 0, 9, 10

Offset: 1

Author

Antti Karttunen, Dec 11 2024

Keywords

• sign
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to prime indices in the factorization of n.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A003961, A016825, A151800, A346234, A378606 (Dirichlet inverse).

Inverse Möbius transform of A349388.

Programs

Mathematica

f[p_, e_] := (p^(e + 1) - NextPrime[p]*(p^e - 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 12 2025 *)

PARI

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A346234(n) = (moebius(n)*A003961(n));
A378607(n) = sumdiv(n,d,sigma(d)*A346234(n/d));

Formula

a(n) = Sum_{d|n} A000203(d)*A346234(n/d).

a(n) = Sum_{d|n} A349388(d).

Multiplicative with a(p^e) = (p^(e+1) - nextprime(p)*(p^e-1) - 1)/(p-1), where nextprime(p) = A151800(p). - Amiram Eldar, Jan 12 2025