A365552 - OEIS

%I #9 Sep 09 2023 06:48:16

%S 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,4,1,1,

%T 1,4,1,1,1,3,1,1,1,2,2,1,1,3,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,4,1,1,1,2,

%U 1,1,1,6,1,1,2,2,1,1,1,3,3,1,1,2,1,1,1

%N The number of exponentially odd divisors of the powerful part of n.

%C First differs from A095691 at n = 512.

%H Amiram Eldar, <a href="/A365552/b365552.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A322483(A057521(n)).

%F Multiplicative with a(p) = 1 and a(p^e) = floor((e+3)/2) for e >= 2.

%F Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^(3*s) - 1/p^(4*s)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 1/p^3 - 1/p^4) = 1.80989829762278336163... .

%t f[p_, e_] := If[e == 1, 1, Floor[(e + 3)/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecprod(apply(x -> if(x == 1, 1, (x+3)\2), factor(n)[, 2]));

%Y Cf. A013661, A095691, A057521, A322483.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Sep 08 2023