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

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

Author

Leroy Quet, Jul 18 2008

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Complement of A383397.

Cf. A020639, A141809, A141810, A126855.

Programs

Mathematica

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

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

Extensions

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

Simpler name from Mike Jones, Jun 15 2022