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

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

Author

Amiram Eldar, Oct 10 2023

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A065463, A268335, A366438.

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

Programs

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]

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

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... .