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A362986
a(n) = A000203(A036966(n)), the sum of divisors of the n-th cubefull number A036966(n).
6
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
1,2
LINKS
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).
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... .
MATHEMATICA
DivisorSigma[1, Select[Range[10^4], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, May 12 2023
STATUS
approved