A370239 - OEIS

%I #11 Feb 13 2024 03:39:46

%S 1,1,1,5,1,1,1,5,10,1,1,5,1,1,1,5,1,10,1,5,1,1,1,5,26,1,10,5,1,1,1,5,

%T 1,1,1,50,1,1,1,5,1,1,1,5,10,1,1,5,50,26,1,5,1,10,1,5,1,1,1,5,1,1,10,

%U 5,1,1,1,5,1,1,1,50,1,1,26,5,1,1,1,5,10,1,1

%N The sum of divisors of n that are squares of squarefree numbers.

%C The number of these divisors is A323308(n).

%H Amiram Eldar, <a href="/A370239/b370239.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) >= 1, with equality if and only if n is squarefree (A005117).

%F a(n) = A071327(n) + 1 if and only if n is not in A036785.

%F Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(4*s-4).

%F Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 2*zeta(3/2)/Pi^2 = 0.5293779248... .

%t f[p_, e_] := If[e == 1, 1, 1 + p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + f[i,1]^2));}

%Y Cf. A005117, A036785, A048250, A062503, A071327, A078434, A323308, A370240.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Feb 13 2024