A097320 - OEIS

A097320

Numbers with more than one prime factor and, in the ordered 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

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

LINKS

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

EXAMPLE

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. Supersequence of A096156.

Cf. A097319, A230766.