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

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.

Cf. A097319, A230766.

Sequence in context: A112769 A360253 A389144 * A332956 A386417 A204825

Adjacent sequences: A097317 A097318 A097319 * A097321 A097322 A097323

Keyword

nonn,easy

Author

Ralf Stephan, Aug 04 2004

Extensions

Edited by Peter Munn, Jun 01 2025

Status

approved