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A366439 The sum of divisors of the exponentially odd numbers (A268335).

%I A366439 #10 Oct 11 2023 18:23:56
%S A366439 1,3,4,6,12,8,15,18,12,14,24,24,18,20,32,36,24,60,42,40,30,72,32,63,
%T A366439 48,54,48,38,60,56,90,42,96,44,72,48,72,54,120,72,120,80,90,60,62,96,
%U A366439 84,144,68,96,144,72,74,114,96,168,80,126,84,108,132,120,180,90
%N A366439 The sum of divisors of the exponentially odd numbers (A268335).
%H A366439 Amiram Eldar, <a href="/A366439/b366439.txt">Table of n, a(n) for n = 1..10000</a>
%F A366439 a(n) = A000203(A268335(n)).
%F A366439 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/(2*d^2)) * Product_{p prime} (1 + 1/(p^5-p)) = 1.045911669131479732932..., where d = 0.7044422... (A065463) is the asymptotic density of the exponentially odd numbers.
%F A366439 The asymptotic mean of the abundancy index of the exponentially odd numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A268335(k) = (1/d) * Product_{p prime} (1 + 1/(p^5-p)) = 2 * c * d = 1.4735686365073812503199... .
%t A366439 f[p_, e_] := (p^(e+1)-1)/(p-1); s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;;, 2]], OddQ], Times @@ f @@@ fct, Nothing]]; s[1] = 1; Array[s, 100]
%o A366439 (PARI) lista(max) = for(k = 1, max, my(f = factor(k), isexpodd = 1); for(i = 1, #f~, if(!(f[i, 2] % 2), isexpodd = 0; break)); if(isexpodd, print1(sigma(f), ", ")));
%o A366439 (Python)
%o A366439 from math import prod
%o A366439 from itertools import count, islice
%o A366439 from sympy import factorint
%o A366439 def A366439_gen(): # generator of terms
%o A366439     for n in count(1):
%o A366439         f = factorint(n)
%o A366439         if all(e&1 for e in f.values()):
%o A366439             yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
%o A366439 A366439_list = list(islice(A366439_gen(),30)) # _Chai Wah Wu_, Oct 11 2023
%Y A366439 Cf. A000203, A065463, A268335, A366438.
%Y A366439 Similar sequences: A062822, A065764, A180114, A362986, A366440.
%K A366439 nonn,easy
%O A366439 1,2
%A A366439 _Amiram Eldar_, Oct 10 2023