%I #10 Oct 11 2023 18:23:56

%S 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 48,54,48,38,60,56,90,42,96,44,72,48,72,54,120,72,120,80,90,60,62,96,

%U 84,144,68,96,144,72,74,114,96,168,80,126,84,108,132,120,180,90

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

%H Amiram Eldar, <a href="/A366439/b366439.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000203(A268335(n)).

%F 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 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 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 (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 (Python)

%o from math import prod

%o from itertools import count, islice

%o from sympy import factorint

%o def A366439_gen(): # generator of terms

%o for n in count(1):

%o f = factorint(n)

%o if all(e&1 for e in f.values()):

%o yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())

%o A366439_list = list(islice(A366439_gen(),30)) # _Chai Wah Wu_, Oct 11 2023

%Y Cf. A000203, A065463, A268335, A366438.

%Y Similar sequences: A062822, A065764, A180114, A362986, A366440.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Oct 10 2023