cp's OEIS Frontend

A365331 The number of divisors of the largest square dividing n.

%I A365331 #16 Sep 02 2023 08:18:38
%S A365331 1,1,1,3,1,1,1,3,3,1,1,3,1,1,1,5,1,3,1,3,1,1,1,3,3,1,3,3,1,1,1,5,1,1,
%T A365331 1,9,1,1,1,3,1,1,1,3,3,1,1,5,3,3,1,3,1,3,1,3,1,1,1,3,1,1,3,7,1,1,1,3,
%U A365331 1,1,1,9,1,1,3,3,1,1,1,5,5,1,1,3,1,1,1
%N A365331 The number of divisors of the largest square dividing n.
%C A365331 All the terms are odd.
%C A365331 The sum of these divisors is A365332(n).
%C A365331 The number of divisors of the square root of the largest square dividing n is A046951(n).
%H A365331 Amiram Eldar, <a href="/A365331/b365331.txt">Table of n, a(n) for n = 1..10000</a>
%F A365331 a(n) = A000005(A008833(n)).
%F A365331 a(n) = 1 if and only if n is squarefree (A005117).
%F A365331 Multiplicative with a(p^e) = e + 1 - (e mod 2).
%F A365331 Dirichlet g.f.: zeta(s)*zeta(2*s)^2/zeta(4*s).
%F A365331 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/2.
%F A365331 More precise asymptotics: Sum_{k=1..n} a(k) ~ 5*n/2 + 3*zeta(1/2)*sqrt(n)/Pi^2 * (log(n) + 4*gamma - 2  - 24*zeta'(2)/Pi^2 + zeta'(1/2)/zeta(1/2)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 02 2023
%p A365331 a:= n-> mul(2*iquo(i[2], 2)+1, i=ifactors(n)[2]):
%p A365331 seq(a(n), n=1..100);  # _Alois P. Heinz_, Sep 01 2023
%t A365331 f[p_, e_] := e + 1 - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365331 (PARI) a(n) = vecprod(apply(x -> x + 1 - x%2, factor(n)[, 2]));
%o A365331 (PARI) a(n) = numdiv(n/core(n)); \\ _Michel Marcus_, Sep 02 2023
%Y A365331 Cf. A000005, A005117, A008833, A046951, A365332.
%K A365331 nonn,easy,mult
%O A365331 1,4
%A A365331 _Amiram Eldar_, Sep 01 2023