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

1,2

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.

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).

Mathematica

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

Prog

(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

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.

Sequence in context: A250222 A026251 A205721 * A085190 A285702 A371064

Adjacent sequences: A378217 A378218 A378219 * A378221 A378222 A378223

Keyword

sign,mult,changed

Author

Antti Karttunen, Nov 23 2024

Status

approved