A230766 -id:A230766 - cp's OEIS frontend

A097318 Numbers with more than one prime factor and, in the ordered factorization, the exponent never increases when read from left to right.

6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114

Offset: 1

Ralf Stephan, Aug 04 2004

nonn

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

These are numbers whose ordered prime signature is weakly decreasing. Weakly increasing is A304678. Ordered prime signature is A124010. - Gus Wiseman, Nov 10 2019

			60 is 2^2*3^1*5^1, A001221(60)=3 and 2>=1>=1, so 60 is in sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
S. Ramanujan, Asymptotic formulas for the distribution of integers of various types, Proc. London Math. Soc. 2, 16 (1917), 112-132.

Subset of A024619. Cf. A097319, A097320, A230766.

Cf. A001222, A025487, A118914, A124010, A304678, A329138, A329142.

q:= n-> (l-> (t-> t>1 and andmap(i-> l[i, 2]>=l[i+1, 2],
        [$1..t-1]))(nops(l)))(sort(ifactors(n)[2])):
select(q, [$1..120])[];  # Alois P. Heinz, Nov 11 2019

fQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Length[f] > 1 && Max[Differences[f]] <= 0]; Select[Range[2, 200], fQ] (* T. D. Noe, Nov 04 2013 *)

for(n=1, 130, F=factor(n); t=0; s=matsize(F)[1]; if(s>1, for(k=1, s-1, if(F[k, 2]

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.

Original entry on oeis.org

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

Ralf Stephan, Aug 04 2004

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

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

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

Subsequence of A126706, A097318, A112769, A304686.
Subsequences: A057715, A096156.
Cf. A097319, A230766.

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 *)

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", "))))

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

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

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).

Edited by Peter Munn, Jun 01 2025

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A097318 Numbers with more than one prime factor and, in the ordered factorization, the exponent never increases when read from left to right.

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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.

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