cp's OEIS Frontend

A368104 The number of bi-unitary divisors of the powerful part of n.

%I A368104 #9 Dec 12 2023 08:01:49
%S A368104 1,1,1,2,1,1,1,4,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,4,2,1,4,2,1,1,1,6,1,1,
%T A368104 1,4,1,1,1,4,1,1,1,2,2,1,1,4,2,2,1,2,1,4,1,4,1,1,1,2,1,1,2,6,1,1,1,2,
%U A368104 1,1,1,8,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1
%N A368104 The number of bi-unitary divisors of the powerful part of n.
%H A368104 Amiram Eldar, <a href="/A368104/b368104.txt">Table of n, a(n) for n = 1..10000</a>
%F A368104 a(n) = A286324(A057521(n)).
%F A368104 Multiplicative with a(p^e) = e if e is even or e = 1, and e + 1 otherwise.
%F A368104 a(n) >= 1, with equality if and only if n is squarefree (A005117).
%F A368104 a(n) <= A286324(n), with equality if and only if n is powerful (A001694).
%F A368104 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 2/p^3 - 1/p^4) = 2.12258268547914758409... .
%t A368104 f[p_, e_] := If[e == 1 || EvenQ[e], e, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368104 (PARI) a(n) = vecprod(apply(x -> if(x == 1 || !(x%2), x, x+1), factor(n)[, 2]));
%Y A368104 Cf. A013661, A001694, A005117, A057521, A286324.
%Y A368104 Similar sequences: A323308, A357669, A368106.
%K A368104 nonn,easy,mult
%O A368104 1,4
%A A368104 _Amiram Eldar_, Dec 12 2023