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A380326 a(n) is the sum of squarefree divisors of the n-th exponentially odd number.

%I A380326 #17 Jan 21 2025 10:35:37
%S A380326 1,3,4,6,12,8,3,18,12,14,24,24,18,20,32,36,24,12,42,4,30,72,32,3,48,
%T A380326 54,48,38,60,56,18,42,96,44,72,48,72,54,12,72,24,80,90,60,62,96,84,
%U A380326 144,68,96,144,72,74,114,96,168,80,126,84,108,132,120,36,90,112
%N A380326 a(n) is the sum of squarefree divisors of the n-th exponentially odd number.
%C A380326 The number of squarefree divisors of the n-th exponentially odd number is A366534(n).
%H A380326 Amiram Eldar, <a href="/A380326/b380326.txt">Table of n, a(n) for n = 1..10000</a>
%F A380326 a(n) = A048250(A268335(n)).
%F A380326 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1/d^2) * Product_{p prime} (1 - 1/(p*(p^2+1))) = 1.73161118511498727822..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
%t A380326 f[p_, e_] := p + 1; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], OddQ], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]
%o A380326 (PARI) A048250(f) = vecprod(apply(x -> x+1, f[, 1]));
%o A380326 list(lim) = for(k = 1, lim, my(f = factor(k), isexpodd = 1); for(i = 1, #f~, if(!(f[i, 2] % 2), isexpodd = 0; break)); if(isexpodd, print1(A048250(f), ", ")));
%Y A380326 Cf. A005117, A048250, A065463, A268335, A366439, A366534, A366535.
%K A380326 nonn,easy
%O A380326 1,2
%A A380326 _Amiram Eldar_, Jan 20 2025