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A367417 The squarefree kernels of the exponentially odd numbers: a(n) = A007947(A268335(n)).

%I A367417 #11 Nov 18 2023 10:42:34
%S A367417 1,2,3,5,6,7,2,10,11,13,14,15,17,19,21,22,23,6,26,3,29,30,31,2,33,34,
%T A367417 35,37,38,39,10,41,42,43,46,47,51,53,6,55,14,57,58,59,61,62,65,66,67,
%U A367417 69,70,71,73,74,77,78,79,82,83,85,86,87,22,89,91,93,94,95
%N A367417 The squarefree kernels of the exponentially odd numbers: a(n) = A007947(A268335(n)).
%H A367417 Amiram Eldar, <a href="/A367417/b367417.txt">Table of n, a(n) for n = 1..10000</a>
%F A367417 a(n) = A367406(n)/A268335(n).
%F A367417 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (zeta(4)/d^2) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 1.25661974314007532208..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
%t A367417 s[n_] := Times @@ FactorInteger[n][[;; , 1]]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]
%o A367417 (PARI) b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1], 0)); }
%o A367417 lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", "))); }
%Y A367417 Cf. A007947, A268335, A367406.
%Y A367417 Cf. A013662, A065463.
%K A367417 nonn,easy
%O A367417 1,2
%A A367417 _Amiram Eldar_, Nov 17 2023