A349385 Dirichlet convolution of A048673 with the Dirichlet inverse of A003961, where A003961 is fully multiplicative with a(p) = nextprime(p), and A048673(n) = (1+A003961(n))/2.

1, -1, -2, -1, -3, 4, -5, -1, -2, 6, -6, 4, -8, 10, 12, -1, -9, 4, -11, 6, 20, 12, -14, 4, -3, 16, -2, 10, -15, -24, -18, -1, 24, 18, 30, 4, -20, 22, 32, 6, -21, -40, -23, 12, 12, 28, -26, 4, -5, 6, 36, 16, -29, 4, 36, 10, 44, 30, -30, -24, -33, 36, 20, -1, 48, -48, -35, 18, 56, -60, -36, 4, -39, 40, 12, 22, 60

Offset: 1

Author

Antti Karttunen, Nov 17 2021

Keywords

• sign

Comments

Convolving this with A003973 gives A336840.

Links

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

Crossrefs

Cf. A003961, A048673, A346234, A349384 (Dirichlet inverse), A349386 (sum with it).

Cf. also A003973, A336840.

Programs

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048673(n) = (A003961(n)+1)/2;
A346234(n) = (moebius(n)*A003961(n));
A349385(n) = sumdiv(n,d,A048673(n/d)*A346234(d));

Formula

a(n) = Sum_{d|n} A048673(n/d) * A346234(d).

a(n) = A349386(n) - A349384(n).