Voozh

A097320

Numbers with more than one distinct prime factor and, in the ordered (canonical) factorization, the exponent always decreases when read from left to right.

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 63, 68, 72, 76, 80, 88, 92, 96, 99, 104, 112, 116, 117, 124, 135, 136, 144, 148, 152, 153, 160, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 244, 248, 261, 268, 272, 275, 279, 284, 288

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OFFSET

1,1

COMMENTS

The numbers in A304686 that are not prime powers. - Peter Munn, Jun 01 2025

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..10000

FORMULA

If n = Product_{k=1..m} p(k)^e(k), with p(k) > p(k-1) for k > 1, then m > 1, e(1) > e(2) > ... > e(m).

EXAMPLE

The ordered (canonical) factorization of 80 is 2^4 * 5^1 and 4 > 1, so 80 is in sequence.

MATHEMATICA

fQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Length[f] > 1 && Max[Differences[f]] < 0]; Select[Range[2, 288], fQ] (* T. D. Noe, Nov 04 2013 *)

PROG

(PARI) for(n=1, 320, F=factor(n); t=0; s=matsize(F)[1]; if(s>1, for(k=1, s-1, if(F[k, 2]<=F[k+1, 2], t=1; break)); if(!t, print1(n", "))))

(PARI) is(n) = my(f = factor(n)[, 2]); #f > 1 && vecsort(f, , 12) == f \\ Rick L. Shepherd, Jan 17 2018

(Python)

from sympy import factorint

def ok(n):

e = list(factorint(n).values())

return 1 < len(e) == len(set(e)) and e == sorted(e, reverse=True)

print([k for k in range(289) if ok(k)]) # Michael S. Branicky, Dec 20 2021

CROSSREFS

Subsequence of A126706, A097318, A112769, A304686.

Subsequences: A057715, A096156.