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%I #17 May 13 2023 06:18:02
%S 1,2,2,3,2,5,3,2,6,7,2,6,3,10,6,11,5,2,6,13,14,10,6,15,3,2,6,17,6,7,
%T 19,14,10,6,21,22,10,2,23,6,5,6,15,26,3,14,10,29,6,30,31,22,6,10,2,33,
%U 15,6,34,35,6,21,11,26,37,14,38,39,14,10,41,6,42,30,43,22,6,10
%N Squarefree kernels of powerful numbers (A001694).
%H Amiram Eldar, <a href="/A084371/b084371.txt">Table of n, a(n) for n = 1..10000</a>
%H Rafael Jakimczuk, <a href="https://doi.org/10.12988/imf.2017.7759">The kernel of powerful numbers</a>, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, Remark 2.5, p. 729.
%F a(n) = A007947(A001694(n)).
%F From _Amiram Eldar_, May 13 2023: (Start)
%F Sum_{a(k) < x} a(k) = (1/2) * x + o(x) (Jakimczuk, 2017).
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/zeta(3/2))^2/2 = 0.1058641473... . (End)
%e A001694(11) = 64 = 2^6 -> a(11) = 2,
%e A001694(12) = 72 = 2^3 * 3^2 -> a(12) = 2*3 = 6,
%e A001694(13) = 81 = 3^4 -> a(13) = 3.
%t s = {1}; Do[f = FactorInteger[n]; If[Min @ f[[;;, 2]] > 1, AppendTo[s, Times @@ f[[;;, 1]]]], {n, 2, 10^4}]; s (* _Amiram Eldar_, Aug 22 2019 *)
%o (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o lista(nn) = apply(x->rad(x), select(x->ispowerful(x), [1..nn])); \\ _Michel Marcus_, Aug 22 2019
%Y Cf. A001694, A007947, A090699.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Jun 23 2003
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