A378220 Dirichlet inverse of phi(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

1, -2, -4, -2, -6, 8, -10, -2, -4, 12, -12, 8, -16, 20, 24, -2, -18, 8, -22, 12, 40, 24, -28, 8, -6, 32, -4, 20, -30, -48, -36, -2, 48, 36, 60, 8, -40, 44, 64, 12, -42, -80, -46, 24, 24, 56, -52, 8, -10, 12, 72, 32, -58, 8, 72, 20, 88, 60, -60, -48, -66, 72, 40, -2, 96, -96, -70, 36, 112, -120, -72, 8, -78, 80, 24

Offset: 1

Author

Antti Karttunen, Nov 23 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.

Crossrefs

Cf. A023900, A003961, A048673, A151800, A349385, A378216.

Dirichlet inverse of A003972.

Inverse Möbius transform of A346234.

After the initial term, A349385 doubled.

Cf. A346246, A378216.

Programs

Mathematica

f[p_, e_] := 1 - NextPrime[p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 2025 *)

PARI

A378220(n) = factorback(apply(p -> 1-nextprime(1+p), factor(n)[, 1]));

Python

from math import prod
from sympy import nextprime, primefactors
def A378220(n): return prod(1-nextprime(p) for p in primefactors(n)) # Chai Wah Wu, Nov 23 2024

Formula

Multiplicative with a(p^e) = (1-q), where q = A151800(p), i.e., the least prime > p.

a(n) = A023900(A003961(n)).

For n > 1, a(n) = 2*A349385(n).

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

a(n) = Sum_{d|n} A346246(d)*A378216(n/d).