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%I #22 Sep 01 2023 04:10:00
%S 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 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 24,9,9,25,12,4,12,3,12,9,9,18,21,3,28,27,36,3,15
%N Number of divisors of the n-th powerful number A001694(n).
%H Amiram Eldar, <a href="/A363194/b363194.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 a(n) = A000005(A001694(n)).
%F Sum_{a(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... .
%F a(n) = A343443(A306458(n)). - _Amiram Eldar_, Sep 01 2023
%t DivisorSigma[0, Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]
%o (PARI) apply(numdiv, select(ispowerful, [1..3000]))
%o (Python)
%o from itertools import count, islice
%o from math import prod
%o from sympy import factorint
%o def A363194_gen(): # generator of terms
%o for n in count(1):
%o f = factorint(n).values()
%o if all(e>1 for e in f):
%o yield prod(e+1 for e in f)
%o A363194_list = list(islice(A363194_gen(),20)) # _Chai Wah Wu_, May 21 2023
%Y Cf. A000005, A001694, A180114, A306458, A343443.
%Y Similar sequences: A072048, A076400, A363195.
%K nonn,easy
%O 1,2
%A _Amiram Eldar_, May 21 2023
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