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%I #18 Nov 30 2023 02:52:44
%S 1,4,9,25,36,49,16,100,121,169,196,225,289,361,441,484,529,144,676,81,
%T 841,900,961,64,1089,1156,1225,1369,1444,1521,400,1681,1764,1849,2116,
%U 2209,2601,2809,324,3025,784,3249,3364,3481,3721,3844,4225,4356,4489,4761
%N The exponentially odd numbers (A268335) multiplied by their squarefree kernels (A007947).
%C Analogous to A355038, with the exponentially odd numbers instead of the square numbers (A000290).
%C This sequence is a permutation of the square numbers.
%H Amiram Eldar, <a href="/A367406/b367406.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A064549(A268335(n)).
%F a(n) = A268335(n)*A367417(n).
%F a(n) = A367407(n)^2.
%F a(n) = A268335(n)^2/A367418(n).
%F Sum_{k=1..n} a(k) = c * n^3 / 3, where c = (Pi^2/(15*d^3)) * Product_{p prime} (1 - 1/(p^3*(p+1))) = 1.78385074227198915372..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
%F a(n) = A053143(A268335(n)). - _Amiram Eldar_, Nov 30 2023
%t s[n_] := n * Times @@ FactorInteger[n][[;;, 1]]; s /@ Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]
%o (PARI) b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, f[i,1]^(f[i,2]+1), 0));}
%o lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", ")));}
%Y Cf. A000290, A007947, A053143, A064549, A065463, A268335, A355038, A367407, A367417, A367418.
%K nonn,easy
%O 1,2
%A _Amiram Eldar_, Nov 17 2023
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