A369309 - OEIS

%I #9 Jan 19 2024 16:56:26

%S 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,

%T 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,

%U 0,0,0,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0

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

%H Amiram Eldar, <a href="/A369309/b369309.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F 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).

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

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

%Y Cf. A001694, A005361, A369310.

%Y Cf. A001620, A002117, A078434, A244115.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Jan 19 2024