|
%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
|
