A362986 a(n) = A000203(A036966(n)), the sum of divisors of the n-th cubefull number A036966(n).

1, 15, 31, 40, 63, 127, 121, 156, 255, 600, 364, 511, 400, 1240, 1023, 781, 1815, 1093, 2520, 2340, 2047, 3751, 1464, 5080, 5460, 4836, 4095, 3280, 2380, 2801, 7623, 6000, 3906, 6240, 10200, 11284, 9828, 8191, 5220, 11715, 15367, 12400, 16395, 9841, 7240, 20440

Offset: 1

Author

Amiram Eldar, May 12 2023

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (7).

Crossrefs

Cf. A000203, A036966, A180114, A362974, A362985.

Programs

Mathematica

DivisorSigma[1, Select[Range[10^4], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]

PARI

lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 2, print1(sigma(k), ", ")));

Python

from itertools import count, islice
from math import prod
from sympy import factorint
def A362986_gen(): # generator of terms
    for n in count(1):
        f = factorint(n)
        if all(e>2 for e in f.values()):
            yield prod((p**(e+1)-1)//(p-1) for p,e in f.items())
A362986_list = list(islice(A362986_gen(),20)) # Chai Wah Wu, May 21 2023

Formula

Sum_{A036966(k) < x} a(k) = c * x^(4/3) + O(x^(113/96 + eps)), where c = A362985 * A362974 / 4 = 2.8912833599... (Jakimczuk and Lalín, 2022). [corrected Sep 21 2024]

Sum_{k=1..n} a(k) ~ c * n^4, where c = A362985 / (4 * A362974^3) = 0.006135085083... .