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A363194 Number of divisors of the n-th powerful number A001694(n).

%I A363194 #26 Sep 21 2024 14:47:16
%S A363194 1,3,4,3,5,3,4,6,9,3,7,12,5,9,12,3,4,8,15,3,9,12,16,9,6,9,18,3,15,4,3,
%T A363194 12,15,20,9,9,12,10,3,21,5,20,12,9,7,15,18,3,24,27,3,12,18,16,11,9,12,
%U A363194 24,9,9,25,12,4,12,3,12,9,9,18,21,3,28,27,36,3,15
%N A363194 Number of divisors of the n-th powerful number A001694(n).
%H A363194 Amiram Eldar, <a href="/A363194/b363194.txt">Table of n, a(n) for n = 1..10000</a>
%H A363194 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 A363194 a(n) = A000005(A001694(n)).
%F A363194 Sum_{A001694(k) < x} a(k) = c_1 * sqrt(x) * log(x)^2 + c_2 * sqrt(x) * log(x) + c_3 * sqrt(x) + O(x^(5/12 + eps)), where c_1, c_2 and c_3 are constants. c_1 = Product_{p prime} (1 + 4/p^(3/2) - 1/p^2 - 6/p^(5/2) + 2/p^(7/2))/8 = 0.516273682988566836609... . [corrected Sep 21 2024]
%F A363194 a(n) = A343443(A306458(n)). - _Amiram Eldar_, Sep 01 2023
%t A363194 DivisorSigma[0, Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]
%o A363194 (PARI) apply(numdiv, select(ispowerful, [1..3000]))
%o A363194 (Python)
%o A363194 from itertools import count, islice
%o A363194 from math import prod
%o A363194 from sympy import factorint
%o A363194 def A363194_gen(): # generator of terms
%o A363194     for n in count(1):
%o A363194         f = factorint(n).values()
%o A363194         if all(e>1 for e in f):
%o A363194             yield prod(e+1 for e in f)
%o A363194 A363194_list = list(islice(A363194_gen(),20)) # _Chai Wah Wu_, May 21 2023
%Y A363194 Cf. A000005, A001694, A180114, A306458, A343443.
%Y A363194 Similar sequences: A072048, A076400, A363195.
%K A363194 nonn,easy
%O A363194 1,2
%A A363194 _Amiram Eldar_, May 21 2023