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A289691
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List of composite numbers n such that the prime decomposition of n contains an initial product of prime powers greater than a later prime, but in which no proper divisor of n is in the sequence.
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1
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12, 30, 40, 45, 56, 63, 70, 105, 154, 165, 175, 176, 182, 195, 198, 208, 220, 231, 234, 260, 273, 275, 286, 297, 306, 325, 340, 351, 357, 374, 380, 385, 399, 418, 425
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OFFSET
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1,1
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COMMENTS
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Definition: Let the prime factorization of n be n = p1^e1*...*pj^ej*p(j+1)^e(j+1)*...*pm^em, with the primes in ascending order and the ej > 0. If in n there exists a partial product p1^e1*...*pk^ek > p(k+1) for some p(k+1) in the factorization, and also such that no proper divisor of n is also in the sequence, then n is in the sequence.
Context: This sequence is a subsequence of A289484, and might be called "A289484 primes," although not primes in the usual sense (nor do they obey a unique factorization law). Every number in A289484 is a multiple of at least one number in this sequence, and if n is in this sequence, then n and every multiple of n is in A289484 although no multiple of n(except n itself) will be in this sequence.
Properties: If n is in the sequence, then no multiple of n is in it, except n itself. No primes or prime powers, nor any composite powers, are in the sequence. A number in the sequence that is squarefree must have at least three prime divisors. Rather than being closed under multiplication, this sequence is anticlosed: No product or power of numbers in it are in the sequence. This causes it to be the minimal sequence that generates A289484. (End)
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LINKS
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EXAMPLE
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30 = 2*3*5 is in the sequence since 2*3 > 5 and no proper divisor of 30 satisfies having an initial product greater than a later prime. Counterexamples: 42 = 2*3*7 is not in the sequence because 2*3 < 7. 60 = 2^2*3*5 is also not in the sequence, because although 2^2 > 3 (and 2^2*3 > 5, for that matter), the proper divisor 30 is already in the sequence.
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MATHEMATICA
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With[{S = Select[Range[10^3], Function[f, AnyTrue[Range[Length@ f - 1], Times @@ Map[#1^#2 & @@ # &, #1] > #2[[1, 1]] & @@ TakeDrop[f, #] &]]@ FactorInteger@ # &]}, Select[#, # > 0 &] &@ Table[Function[{d, n}, First[n] Boole@ NoneTrue[d, Divisible[First@ n, #] &]] @@ TakeDrop[Take[S, n], n - 1], {n, Length@ S}]] (* Michael De Vlieger, Aug 17 2017 *)
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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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