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A363195 Number of divisors of the n-th cubefull number A036966(n).

%I A363195 #24 Sep 21 2024 14:46:41
%S A363195 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,
%T A363195 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,
%U A363195 16,24,35,4,40,5,42,7,32,15,6,20,32,16,20,10,30,45,20
%N A363195 Number of divisors of the n-th cubefull number A036966(n).
%H A363195 Amiram Eldar, <a href="/A363195/b363195.txt">Table of n, a(n) for n = 1..10000</a>
%H A363195 Rafael Jakimczuk and Matilde Lalín, <a href="https://doi.org/10.7546/nntdm.2022.28.4.617-634">Asymptotics of sums of divisor functions over sequences with restricted factorization structure</a>, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (8).
%F A363195 a(n) = A000005(A036966(n)).
%F A363195 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]
%t A363195 DivisorSigma[0, Select[Range[25000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 2 &]]
%o A363195 (PARI) lista(kmax) = for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 2, print1(numdiv(k), ", ")));
%o A363195 (Python)
%o A363195 from itertools import count, islice
%o A363195 from math import prod
%o A363195 from sympy import factorint
%o A363195 def A363195_gen(): # generator of terms
%o A363195     for n in count(1):
%o A363195         f = factorint(n).values()
%o A363195         if all(e>2 for e in f):
%o A363195             yield prod(e+1 for e in f)
%o A363195 A363195_list = list(islice(A363195_gen(),20)) # _Chai Wah Wu_, May 21 2023
%Y A363195 Cf. A000005, A036966, A362986.
%Y A363195 Similar sequences: A072048, A076400, A363194.
%K A363195 nonn,easy
%O A363195 1,2
%A A363195 _Amiram Eldar_, May 21 2023