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

1,2

Links

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

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

Cf. A000203, A065463, A268335, A366438.

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

Sequence in context: A366535 A368469 A374457 * A062822 A175894 A175029

Adjacent sequences: A366436 A366437 A366438 * A366440 A366441 A366442

Keyword

nonn,easy

Author

Amiram Eldar, Oct 10 2023

Status

approved