A369309 The number of powerful divisors d of n such that n/d is also powerful.

1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^2) = 2 and a(p^e) = e-1 if e != 2.

a(n) > 0 if and only if n is powerful (A001694).

Dirichlet g.f.: (zeta(2*s)*zeta(3*s)/zeta(6*s))^2.

Sum_{k=1..n} a(k) ~ (zeta(3/2)^2/(2*zeta(3)^2)) * sqrt(n) * (log(n) + 4*gamma - 2 + 6*zeta'(3/2)/zeta(3/2) - 12*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).

Mathematica

f[p_, e_] := If[e == 2, 2, e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecprod(apply(x -> if(x==2, 2, x-1), factor(n)[, 2]));

Crossrefs

Cf. A001694, A005361, A369310.

Cf. A001620, A002117, A078434, A244115.

Sequence in context: A100699 A108921 A071548 * A216228 A372602 A374202

Adjacent sequences: A369306 A369307 A369308 * A369310 A369311 A369312

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 19 2024

Status

approved