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A365682 The sum of exponentially squarefree divisors of n.

%I A365682 #9 Sep 16 2023 02:21:42
%S A365682 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,15,18,39,20,42,32,36,24,60,31,
%T A365682 42,40,56,30,72,32,47,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,60,
%U A365682 57,93,72,98,54,120,72,120,80,90,60,168,62,96,104,111,84,144
%N A365682 The sum of exponentially squarefree divisors of n.
%C A365682 The sum of divisors of n that are exponentially squarefree numbers (A209061), i.e., numbers having only squarefree exponents in their canonical prime factorization.
%C A365682 The number of these divisors is A365680(n) and the largest of them is A365683(n).
%H A365682 Amiram Eldar, <a href="/A365682/b365682.txt">Table of n, a(n) for n = 1..10000</a>
%F A365682 Multiplicative with a(p^e) = 1 + Sum_{k=1..e, k squarefree} p^k.
%F A365682 a(n) <= A000203(n), with equality if and only if n is a biquadratefree number (A046100).
%t A365682 f[p_, e_] := 1 + Sum[If[SquareFreeQ[k], p^k, 0], {k, 1, e}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365682 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 1, f[i,2], issquarefree(k) * f[i,1]^k));}
%Y A365682 Cf. A000203, A046100, A209061, A365680, A365683.
%K A365682 nonn,easy,mult
%O A365682 1,2
%A A365682 _Amiram Eldar_, Sep 15 2023