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A386471 The sum of the divisors of n whose exponents in their prime factorization are squares.

%I A386471 #7 Jul 23 2025 07:21:18
%S A386471 1,3,4,3,6,12,8,3,4,18,12,12,14,24,24,19,18,12,20,18,32,36,24,12,6,42,
%T A386471 4,24,30,72,32,19,48,54,48,12,38,60,56,18,42,96,44,36,24,72,48,76,8,
%U A386471 18,72,42,54,12,72,24,80,90,60,72,62,96,32,19,84,144,68,54
%N A386471 The sum of the divisors of n whose exponents in their prime factorization are squares.
%C A386471 The sum of the terms in A197680 that divide n.
%C A386471 The number of these divisors is A386470(n) and the largest of them is A386469(n).
%H A386471 Amiram Eldar, <a href="/A386471/b386471.txt">Table of n, a(n) for n = 1..10000</a>
%F A386471 Multiplicative with a(p^e) = Sum_{k=0..floor(sqrt(e))} p^(k^2).
%F A386471 a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
%F A386471 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + Sum_{k>=2} Sum_{i=(2*k)^2..(2*k+1)^2-1} (-1/p)^i) = 1.05459216969289486594... .
%t A386471 f[p_, e_] := Sum[p^(k^2), {k, 0, Floor[Sqrt[e]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A386471 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, sum(k = 0, sqrtint(f[i,2]), f[i,1]^(k^2)));}
%Y A386471 Cf. A000203, A005117, A197680, A386469, A386470.
%K A386471 nonn,easy,mult
%O A386471 1,2
%A A386471 _Amiram Eldar_, Jul 23 2025