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A183094 a(n) = number of powerful divisors d (excluding 1) of n.

%I A183094 #28 Jul 25 2024 04:29:12
%S A183094 0,0,0,1,0,0,0,2,1,0,0,1,0,0,0,3,0,1,0,1,0,0,0,2,1,0,2,1,0,0,0,4,0,0,
%T A183094 0,3,0,0,0,2,0,0,0,1,1,0,0,3,1,1,0,1,0,2,0,2,0,0,0,1,0,0,1,5,0,0,0,1,
%U A183094 0,0,0,5,0,0,1,1,0,0,0,3,3,0,0,1,0,0,0,2,0,1,0,1,0,0,0,4,0,1,1,3,0,0,0,2,0
%N A183094 a(n) = number of powerful divisors d (excluding 1) of n.
%C A183094 a(n) = number of divisors d of n from set A001694(m) - powerful numbers for m >=2.
%H A183094 Robert Israel, <a href="/A183094/b183094.txt">Table of n, a(n) for n = 1..10000</a>
%H A183094 D. Suryanarayana and R. Sitaramachandra Rao, <a href="http://dx.doi.org/10.1090/S0002-9939-1972-0291104-8">The number of square-full divisors of an integer</a>, Proc. Amer. Math. Soc. 34 (1972), 79-80.
%F A183094 a(n) = A000005(n) - A183095(n) = A005361(n) - 1.
%F A183094 a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = k-1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
%F A183094 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)*zeta(3)/zeta(6) - 1 = A082695 - 1 = 0.9435964368... . - _Amiram Eldar_, Jul 30 2022
%e A183094 For n = 12, set of such divisors is {4}; a(12) = 1.
%p A183094 f:=  n -> convert(map(t->t[2], ifactors(n)[2]),`*`) - 1; # _Robert Israel_, Jul 14 2014
%t A183094 powerfulQ[n_] := Min[ Last@# & /@ FactorInteger[n]] > 1; f[n_] := Length@ Select[ Divisors@ n, powerfulQ]; Array[f, 105] (* _Robert G. Wilson v_, Jul 14 2014 *)
%Y A183094 Cf. A000005, A001694, A005361, A082695, A183095.
%K A183094 nonn
%O A183094 1,8
%A A183094 _Jaroslav Krizek_, Dec 25 2010