cp's OEIS Frontend

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.

%I A097320 #36 Jun 02 2025 15:15:21
%S A097320 12,20,24,28,40,44,45,48,52,56,63,68,72,76,80,88,92,96,99,104,112,116,
%T A097320 117,124,135,136,144,148,152,153,160,164,171,172,175,176,184,188,189,
%U A097320 192,200,207,208,212,224,232,236,244,248,261,268,272,275,279,284,288
%N 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.
%C A097320 The numbers in A304686 that are not prime powers. - _Peter Munn_, Jun 01 2025
%H A097320 Michael S. Branicky, <a href="/A097320/b097320.txt">Table of n, a(n) for n = 1..10000</a>
%F A097320 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).
%e A097320 The ordered (canonical) factorization of 80 is 2^4 * 5^1 and 4 > 1, so 80 is in sequence.
%t A097320 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 *)
%o A097320 (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", "))))
%o A097320 (PARI) is(n) = my(f = factor(n)[,2]); #f > 1 && vecsort(f,,12) == f \\ _Rick L. Shepherd_, Jan 17 2018
%o A097320 (Python)
%o A097320 from sympy import factorint
%o A097320 def ok(n):
%o A097320     e = list(factorint(n).values())
%o A097320     return 1 < len(e) == len(set(e)) and e == sorted(e, reverse=True)
%o A097320 print([k for k in range(289) if ok(k)]) # _Michael S. Branicky_, Dec 20 2021
%Y A097320 Subsequence of A126706, A097318, A112769, A304686.
%Y A097320 Subsequences: A057715, A096156.
%Y A097320 Cf. A097319, A230766.
%K A097320 nonn,easy
%O A097320 1,1
%A A097320 _Ralf Stephan_, Aug 04 2004
%E A097320 Edited by _Peter Munn_, Jun 01 2025