A126855 -id:A126855 - cp's OEIS frontend

A102749 Numbers k such that the largest prime-power dividing k is not a power of the largest prime dividing k.

12, 24, 40, 45, 48, 56, 63, 80, 90, 96, 112, 120, 126, 135, 144, 160, 168, 175, 176, 180, 189, 192, 208, 224, 240, 252, 270, 275, 280, 288, 297, 315, 320, 325, 336, 350, 351, 352, 360, 378, 384, 405, 416, 425, 448, 459, 475, 480, 504, 513, 525, 528, 539, 540

Offset: 1

Leroy Quet, Feb 09 2005

nonn

Does this sequence have finite density? - Franklin T. Adams-Watters, Aug 29 2006

The numbers of terms not exceeding 10^k, for k=1,2,..., are 0, 10, 97, 706, 4779, 31249, 203799, 1322874, 8622492, 56559400, ... Apparently this sequence has an asymptotic density 0. - Amiram Eldar, Mar 20 2021

			45 is a term because 45 = 3^2*5 and 9 (the largest prime-power dividing 45) is not a power of 5 (the largest prime dividing 45).
144 is a term because its largest prime divisor is 3, but the largest prime power divisor, 16, is not a power of 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006530, A034699, A126855.

fQ[n_] := Block[{p = Power @@@ FactorInteger[n]},Last[p] != Max[p]];Select[Range[540], fQ] (* Ray Chandler, May 11 2007 *)

More terms from Franklin T. Adams-Watters, Aug 29 2006

A140831 Numbers in whose canonical prime factorization the powers of the primes do not form an increasing sequence.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Jul 18 2008

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
If k is in the sequence then k*A020639(k)^m is in the sequence for m >= 0. - David A. Corneth, Jun 16 2022
Conjecture: There are infinitely many terms k such that k+1 is also a term. - Mike Jones, Jun 18 2022

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000

Complement of A383397.

Cf. A020639, A141809, A141810, A126855.

fQ[n_] := Block[{f = First@# ^ Last@# & /@ FactorInteger@n}, f != Sort@f]; Select[ Range@ 407, fQ@# &] (* Robert G. Wilson v, Aug 31 2008 *)

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

More terms from Robert G. Wilson v, Aug 31 2008

Simpler name from Mike Jones, Jun 15 2022

cp's OEIS Frontend

A102749 Numbers k such that the largest prime-power dividing k is not a power of the largest prime dividing k.

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