cp's OEIS Frontend

A368105 The number of bi-unitary divisors of n that are powerful (A001694).

%I A368105 #11 Dec 12 2023 08:18:54
%S A368105 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,5,1,1,
%T A368105 1,4,1,1,1,3,1,1,1,2,2,1,1,3,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,5,1,1,1,2,
%U A368105 1,1,1,6,1,1,2,2,1,1,1,3,3,1,1,2,1,1,1
%N A368105 The number of bi-unitary divisors of n that are powerful (A001694).
%C A368105 First differs from A095691 and A365552 at n = 32.
%H A368105 Amiram Eldar, <a href="/A368105/b368105.txt">Table of n, a(n) for n = 1..10000</a>
%F A368105 Multiplicative with a(p^e) = e if e = 2 or e is odd, and e-1 otherwise.
%F A368105 a(n) >= 1, with equality if and only if n is squarefree (A005117).
%F A368105 a(n) <= A286324(n), with equality if and only if n equals the square of a squarefree number (A062503).
%F A368105 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/p^5) = 1.87133814920590891161... .
%t A368105 f[p_, e_] := If[e == 2 || OddQ[e], e, e -1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368105 (PARI) a(n) = vecprod(apply(x -> if(x%2 || x == 2, x, x-1), factor(n)[, 2]));
%Y A368105 Cf. A013661, A001694, A005117, A062503, A286324,
%Y A368105 Similar sequences: A005361, A323308, A360721.
%Y A368105 Cf. A095691, A365552.
%K A368105 nonn,easy,mult
%O A368105 1,4
%A A368105 _Amiram Eldar_, Dec 12 2023