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A366761
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Numbers that have at least one exponent in their canonical prime factorization that is divisible by 3.
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2
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8, 24, 27, 40, 54, 56, 64, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 192, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 320, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 432, 440, 448, 456, 459, 472, 488, 500, 504, 512, 513, 520
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OFFSET
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1,1
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COMMENTS
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Each term has a unique representation of as product of two numbers: one is a cube (A000578) and the second is a number that is not in this sequence.
The asymptotic density of this sequence is 1 - zeta(3) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.10483363599014046584... .
The complement of this sequence is the sequence of numbers called "unitarily 3-free", or "3-skew", by Cohen (1961).
He proved that the asymptotic density of unitarily k-free, i.e., numbers whose prime factorization contain no exponent that is divisible by k, is zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(k+1)) (see p. 228, eq. 3.18). (End)
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n)^s = zeta(s) * (1 - zeta(3*s) * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s))), for s > 1.
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MATHEMATICA
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q[n_] := ! AllTrue[FactorInteger[n][[;; , 2]], ! Divisible[#, 3] &]; Select[Range[500], q]
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PROG
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(PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(!(f[i, 2]%3), return(1))); 0; }
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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