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A080259
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Numbers whose squarefree kernel is not a primorial number, i.e., A007947(a(n)) is not in A002110.
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13
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3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87
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OFFSET
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1,1
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COMMENTS
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Odd prime power p^m, m >= 1 is in the sequence since its squarefree kernel p is odd and not a primorial. Therefore 3^3, 5^2, etc. are in the sequence.
Odd squarefree composite k is in the sequence since its squarefree kernel is odd and thus not a primorial. Therefore 15 and 33 are in the sequence.
Numbers k such that A053669(k) < A006530(k) are in the sequence since the condition A053669(k) < A006530(k) implies the squarefree kernel is not a primorial, etc. (End)
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LINKS
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FORMULA
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EXAMPLE
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1 is not in the sequence because its squarefree kernel is 1, the product of the 0 primes that divide 1 (the "empty product") and therefore the same as A002110(0), the 0-th primorial.
2 is not in the sequence since its squarefree kernel is 2, the smallest prime, hence the same as A002110(1) = 2.
4 is not in the sequence since its squarefree kernel is 2 = A002110(1).
(End)
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MATHEMATICA
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Select[Range[120], Nor[IntegerQ@ Log2[#], And[EvenQ[#], Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}]] &] (* Michael De Vlieger, Jan 23 2024 *)
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PROG
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(PARI) is(n) = {my(f=factor(n)[, 1]); n>1&&primepi(f[#f])>#f} \\ David A. Corneth, May 22 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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