cp's OEIS Frontend

A367989 The sum of square divisors of the largest unitary divisor of n that is a square.

%I A367989 #8 Dec 07 2023 10:26:01
%S A367989 1,1,1,5,1,1,1,1,10,1,1,5,1,1,1,21,1,10,1,5,1,1,1,1,26,1,1,5,1,1,1,1,
%T A367989 1,1,1,50,1,1,1,1,1,1,1,5,10,1,1,21,50,26,1,5,1,1,1,1,1,1,1,5,1,1,10,
%U A367989 85,1,1,1,5,1,1,1,10,1,1,26,5,1,1,1,21,91,1
%N A367989 The sum of square divisors of the largest unitary divisor of n that is a square.
%H A367989 Amiram Eldar, <a href="/A367989/b367989.txt">Table of n, a(n) for n = 1..10000</a>
%F A367989 a(n) = A035316(A350388(n)).
%F A367989 Multiplicative with a(p^e) = (p^(e+2)-1)/(p^2-1) if e is even and 1 otherwise.
%F A367989 a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
%F A367989 Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-2)).
%F A367989 Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) - 1/p^(5/2)) = 0.69451968056653021193... .
%t A367989 f[p_, e_] := If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A367989 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, 1, (f[i,1]^(f[i,2] + 2) - 1)/(f[i,1]^2 - 1)));}
%Y A367989 Cf. A002117, A035316, A268335, A350388, A367988.
%K A367989 nonn,easy,mult
%O A367989 1,4
%A A367989 _Amiram Eldar_, Dec 07 2023