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A140831
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Numbers in whose canonical prime factorization the powers of the primes do not form an increasing sequence.
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1
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12, 24, 40, 45, 48, 56, 60, 63, 80, 84, 90, 96, 112, 120, 126, 132, 135, 144, 156, 160, 168, 175, 176, 180, 189, 192, 204, 208, 224, 228, 240, 252, 264, 270, 275, 276, 280, 288, 297, 300, 312, 315, 320, 325, 336, 348, 350, 351, 352, 360, 372, 378, 384, 405
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OFFSET
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1,1
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COMMENTS
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Previous name was: Let p^b(n,p) be the largest power of the prime p that divides n. The integer n is included if the list of p^b(n,p)'s, where each p is a distinct prime divisor of n, arranged by size of each p^b(n,p) is not in the same order as the list of p^b(n,p)'s arranged by size of each prime p.
This sequence contains no squarefree integers.
90 is the smallest integer in this sequence but not in sequence A126855.
The number of terms < 10^n: 0, 12, 151, 1575, 16154, 161630, 1617052, ..., . - Robert G. Wilson v, Aug 31 2008
If k is in the sequence, then all powers of k are in the sequence. - Mike Jones, Jun 16 2022
Conjecture: There are infinitely many terms k such that k+1 is also a term. - Mike Jones, Jun 18 2022
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LINKS
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EXAMPLE
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The prime factorization of 90 is, when arranged by size of the distinct primes, 2^1 * 3^2 * 5^1. Since 3^2 is > 5^1, even though 5 > 3, 90 is in the sequence.
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MATHEMATICA
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fQ[n_] := Block[{f = First@# ^ Last@# & /@ FactorInteger@n}, f != Sort@f]; Select[ Range@ 407, fQ@# &] (* Robert G. Wilson v, Aug 31 2008 *)
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PROG
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(PARI) is(n) = { my(f = factor(n)); for(i = 1, #f~-1, if(f[i, 1]^f[i, 2] > f[i+1, 1]^f[i+1, 2], return(1) ) ); 0 } \\ David A. Corneth, Jun 16 2022
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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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EXTENSIONS
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STATUS
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approved
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