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A366439
The sum of divisors of the exponentially odd numbers (A268335).
8
1, 3, 4, 6, 12, 8, 15, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 60, 42, 40, 30, 72, 32, 63, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 72, 48, 72, 54, 120, 72, 120, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 180, 90
OFFSET
1,2
LINKS
FORMULA
a(n) = A000203(A268335(n)).
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.
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... .
MATHEMATICA
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]
PROG
(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), ", ")));
(Python)
from math import prod
from itertools import count, islice
from sympy import factorint
def A366439_gen(): # generator of terms
for n in count(1):
f = factorint(n)
if all(e&1 for e in f.values()):
yield prod((p**(e+1)-1)//(p-1) for p, e in f.items())
A366439_list = list(islice(A366439_gen(), 30)) # Chai Wah Wu, Oct 11 2023
CROSSREFS
Similar sequences: A062822, A065764, A180114, A362986, A366440.
Sequence in context: A366535 A368469 A374457 * A062822 A175894 A175029
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Oct 10 2023
STATUS
approved