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%I #46 Dec 01 2023 15:53:01
%S 1,8,27,32,64,125,216,243,343,512,729,864,1000,1024,1331,1728,1944,
%T 2197,2744,3125,3375,4000,4096,4913,5832,6859,7776,8000,9261,10648,
%U 10976,12167,13824,15552,15625,16807,17576,19683,21952,23328,24389,25000,27000,27648,29791
%N Exponentially evil numbers.
%C Or the numbers whose prime power factorization contains primes only in evil exponents (A001969): 0, 3, 5, 6, 9, 10, 12, ...
%C If n is in the sequence, then n^2 is also in the sequence.
%C A268385 maps each term of this sequence to a unique nonzero square (A000290), and vice versa. - _Antti Karttunen_, May 26 2016
%H Amiram Eldar, <a href="/A262675/b262675.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Reinhard Zumkeller)
%F Product_{k=1..A001221(n)} A010059(A124010(n,k)) = 1. - _Reinhard Zumkeller_, Oct 25 2015
%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=2} 1/p^A001969(k)) = Product_{p prime} f(1/p) = 1.2413599378..., where f(x) = (1/(1-x) + Product_{k>=0} (1 - x^(2^k)))/2. - _Amiram Eldar_, May 18 2023, Dec 01 2023
%e 864 = 2^5*3^3; since 5 and 3 are evil numbers, 864 is in the sequence.
%t {1}~Join~Select[Range@ 30000, AllTrue[Last /@ FactorInteger[#], EvenQ@ First@ DigitCount[#, 2] &] &] (* _Michael De Vlieger_, Sep 27 2015, Version 10 *)
%t expEvilQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], EvenQ[DigitCount[#, 2, 1]] &]; With[{max = 30000}, Select[Union[Flatten[Table[i^2*j^3, {j, Surd[max, 3]}, {i, Sqrt[max/j^3]}]]], expEvilQ]] (* _Amiram Eldar_, Dec 01 2023 *)
%o (PARI) isok(n) = {my(f = factor(n)); for (i=1, #f~, if (hammingweight(f[i,2]) % 2, return (0));); return (1);} \\ _Michel Marcus_, Sep 27 2015
%o (Haskell)
%o a262675 n = a262675_list !! (n-1)
%o a262675_list = filter
%o (all (== 1) . map (a010059 . fromIntegral) . a124010_row) [1..]
%o -- _Reinhard Zumkeller_, Oct 25 2015
%o (Perl) use ntheory ":all"; sub isok { my @f = factor_exp($_[0]); return scalar(grep { !(hammingweight($_->[1]) % 2) } @f) == @f; } # _Dana Jacobsen_, Oct 26 2015
%Y Subsequence of A036966.
%Y Apart from 1, a subsequence of A270421.
%Y Indices of ones in A270418.
%Y Sequence A270437 sorted into ascending order.
%Y Cf. A001969, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A010059, A124010, A268385, A270428.
%K nonn,base
%O 1,2
%A _Vladimir Shevelev_, Sep 27 2015
%E More terms from _Michel Marcus_, Sep 27 2015
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