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A337050
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Numbers without an exponent 2 in their prime factorization.
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12
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1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87
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OFFSET
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1,2
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COMMENTS
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Numbers k such that the powerful part (A057521) of k is a cubefull number (A036966).
The asymptotic density of this sequence is Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).
A304364 is apparently a subsequence.
These numbers were named semi-2-free integers by Suryanarayana (1971). - Amiram Eldar, Dec 29 2020
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n)^s = zeta(s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)), for s > 1. - Amiram Eldar, Oct 21 2023
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EXAMPLE
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6 = 2^1 * 3^1 is a term since none of the exponents in its prime factorization is equal to 2.
9 = 3^2 is not a term since it has an exponent 2 in its prime factorization.
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MAPLE
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q:= n-> andmap(i-> i[2]<>2, ifactors(n)[2]):
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MATHEMATICA
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Select[Range[100], !MemberQ[FactorInteger[#][[;; , 2]], 2] &]
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PROG
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(PARI) is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 2, return(0))); 1; } \\ Amiram Eldar, Oct 21 2023
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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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