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A363195
Number of divisors of the n-th cubefull number A036966(n).
5
1, 4, 5, 4, 6, 7, 5, 4, 8, 16, 6, 9, 4, 20, 10, 5, 20, 7, 24, 16, 11, 25, 4, 28, 24, 20, 12, 8, 4, 5, 30, 16, 6, 16, 32, 30, 24, 13, 4, 20, 35, 20, 28, 9, 4, 36, 36, 28, 14, 16, 25, 20, 40, 16, 24, 35, 4, 40, 5, 42, 7, 32, 15, 6, 20, 32, 16, 20, 10, 30, 45, 20
OFFSET
1,2
LINKS
Rafael Jakimczuk and Matilde LalĂ­n, 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. (8).
FORMULA
a(n) = A000005(A036966(n)).
Sum_{A036966(k) < x} a(k) = c_1 * x^(1/3) * log(x)^3 + c_2 * x^(1/3) * log(x)^2 + c_3 * x^(1/3) * log(x) + c_4 * x^(1/3) + O(x^(7/24 + eps)), where c_1, c_2, c_3 and c_4 are constants. c_1 = Product_{p prime} ((1-1/p)^4 * (1 + 1/((p^(1/3) - 1)^2*p^(1/3)) + 3/(p-p^(2/3))))/162 = 0.1346652397135839416... . [corrected Sep 21 2024]
MATHEMATICA
DivisorSigma[0, Select[Range[25000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]
PROG
(PARI) lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 2, print1(numdiv(k), ", ")));
(Python)
from itertools import count, islice
from math import prod
from sympy import factorint
def A363195_gen(): # generator of terms
for n in count(1):
f = factorint(n).values()
if all(e>2 for e in f):
yield prod(e+1 for e in f)
A363195_list = list(islice(A363195_gen(), 20)) # Chai Wah Wu, May 21 2023
CROSSREFS
Similar sequences: A072048, A076400, A363194.
Sequence in context: A303275 A198817 A248624 * A085428 A053025 A010664
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, May 21 2023
STATUS
approved