A349388 Dirichlet convolution of A000027 with A346234 (Dirichlet inverse of A003961), where A003961 is fully multiplicative with a(p) = nextprime(p).

1, -1, -2, -2, -2, 2, -4, -4, -6, 2, -2, 4, -4, 4, 4, -8, -2, 6, -4, 4, 8, 2, -6, 8, -10, 4, -18, 8, -2, -4, -6, -16, 4, 2, 8, 12, -4, 4, 8, 8, -2, -8, -4, 4, 12, 6, -6, 16, -28, 10, 4, 8, -6, 18, 4, 16, 8, 2, -2, -8, -6, 6, 24, -32, 8, -4, -4, 4, 12, -8, -2, 24, -6, 4, 20, 8, 8, -8, -4, 16, -54, 2, -6, -16, 4, 4, 4

Offset: 1

Author

Antti Karttunen, Nov 17 2021

Keywords

• sign
• mult

Comments

Multiplicative because A000027 and A346234 are.

Links

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

Crossrefs

Cf. A000027, A000040, A001223, A003961, A151800, A346234, A349387 (Dirichlet inverse), A349389 (sum with it), A378607 (inverse Möbius transform).

Cf. also A347238.

Programs

Mathematica

f[p_, e_] := p^e - NextPrime[p] * p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

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));
A349388(n) = sumdiv(n,d,d*A346234(n/d));

Formula

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

For all n >= 1, a(A000040(n)) = -A001223(n).

Multiplicative with a(p^e) = p^e - nextprime(p) * p^(e-1), where nextprime function is A151800. - Amiram Eldar, Nov 18 2021