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A262675
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Exponentially evil numbers.
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23
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1, 8, 27, 32, 64, 125, 216, 243, 343, 512, 729, 864, 1000, 1024, 1331, 1728, 1944, 2197, 2744, 3125, 3375, 4000, 4096, 4913, 5832, 6859, 7776, 8000, 9261, 10648, 10976, 12167, 13824, 15552, 15625, 16807, 17576, 19683, 21952, 23328, 24389, 25000, 27000, 27648, 29791
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OFFSET
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1,2
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COMMENTS
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Or the numbers whose prime power factorization contains primes only in evil exponents (A001969): 0, 3, 5, 6, 9, 10, 12, ...
If n is in the sequence, then n^2 is also in the sequence.
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LINKS
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FORMULA
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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
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EXAMPLE
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864 = 2^5*3^3; since 5 and 3 are evil numbers, 864 is in the sequence.
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MATHEMATICA
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{1}~Join~Select[Range@ 30000, AllTrue[Last /@ FactorInteger[#], EvenQ@ First@ DigitCount[#, 2] &] &] (* Michael De Vlieger, Sep 27 2015, Version 10 *)
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 *)
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PROG
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(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
(Haskell)
a262675 n = a262675_list !! (n-1)
a262675_list = filter
(all (== 1) . map (a010059 . fromIntegral) . a124010_row) [1..]
(Perl) use ntheory ":all"; sub isok { my @f = factor_exp($_[0]); return scalar(grep { !(hammingweight($_->[1]) % 2) } @f) == @f; } # Dana Jacobsen, Oct 26 2015
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CROSSREFS
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Apart from 1, a subsequence of A270421.
Sequence A270437 sorted into ascending order.
Cf. A001969, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A010059, A124010, A268385, A270428.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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