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A262675 Exponentially evil numbers. 23
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
A268385 maps each term of this sequence to a unique nonzero square (A000290), and vice versa. - Antti Karttunen, May 26 2016
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
FORMULA
Product_{k=1..A001221(n)} A010059(A124010(n,k)) = 1. - Reinhard Zumkeller, Oct 25 2015
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
EXAMPLE
864 = 2^5*3^3; since 5 and 3 are evil numbers, 864 is in the sequence.
MATHEMATICA
{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 *)
PROG
(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..]
-- Reinhard Zumkeller, Oct 25 2015
(Perl) use ntheory ":all"; sub isok { my @f = factor_exp($_[0]); return scalar(grep { !(hammingweight($_->[1]) % 2) } @f) == @f; } # Dana Jacobsen, Oct 26 2015
CROSSREFS
Subsequence of A036966.
Apart from 1, a subsequence of A270421.
Indices of ones in A270418.
Sequence A270437 sorted into ascending order.
Sequence in context: A056729 A070265 A354179 * A102834 A370786 A116002
KEYWORD
nonn,base
AUTHOR
Vladimir Shevelev, Sep 27 2015
EXTENSIONS
More terms from Michel Marcus, Sep 27 2015
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)