A349379 Möbius transform of A057521 (powerful part of n).

1, 0, 0, 3, 0, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 18, 0, 0, 0, 0, 16, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0

Offset

1,4

Comments

Multiplicative with a(p^e) = 0 if e = 1, p^2 - 1 if e = 2 and p^e - p^(e-1) otherwise. - Amiram Eldar, Nov 18 2021

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = Sum_{d|n} A008683(n/d) * A057521(d).

a(n) = Sum_{d|n} A000010(n/d) * A349441(d).

Mathematica

f[p_, e_] := Which[e > 2, p^e - p^(e - 1), e == 2, p^2 - 1, e == 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

Prog

(PARI)
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A349379(n) = sumdiv(n, d, moebius(n/d)*A057521(d));
(Python)
from math import prod
from sympy import factorint
def A349379(n): return prod(0 if e==1 else p**e - (1 if e==2 else p**(e-1)) for p, e in factorint(n).items()) # Chai Wah Wu, Nov 14 2022

Crossrefs

Cf. A000010, A008683, A057521, A349441.

Cf. also A300717.

Sequence in context: A122480 A096133 A378036 * A293381 A118112 A245552

Adjacent sequences: A349376 A349377 A349378 * A349380 A349381 A349382

Keyword

nonn,mult

Author

Antti Karttunen, Nov 18 2021

Status

approved