A057715 -id:A057715 - 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.

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

A102308 If n = product{primes p(k)|n} p(k)^b(n,p(k)), where p(k) is the k-th prime that divides n (when these primes are listed from smallest to largest) and each b(n,p(k)) is a positive integer, then the sequence contains the non-prime-powers n such that p(k)^b(n,p(k)) > p(k+1) for all k, 1<=k<= -1 + number of distinct prime divisors of n.

Original entry on oeis.org

12, 24, 36, 40, 45, 48, 56, 63, 72, 80, 96, 108, 112, 135, 144, 160, 175, 176, 180, 189, 192, 200, 208, 216, 224, 225, 252, 275, 288, 297, 320, 324, 325, 351, 352, 360, 384, 392, 400, 405, 416, 425, 432, 441, 448, 459, 475, 504, 513, 539, 540, 544, 567, 575

Offset: 1

Leroy Quet, Sep 04 2008

nonn

			252 is factored as 2^2 * 3^2 * 7^1. Since 2^2 > 3 and 3^2 > 7, then 252 is in the sequence. On the other hand, 60 is factored as 2^2 * 3^1 * 5^1. Even though 2^2 > 3, 3^1 is not > 5. So 60 is not in the sequence.

Cf. A057715, A143907.

isok(n) = {my(f = factor(n)); if (#f~ == 1, return (0)); for (i=1, #f~ - 1, if (f[i, 1]^f[i, 2] <= f[i+1, 1], return (0));); return (1);} \\ Michel Marcus, Jan 19 2014

Extended by Ray Chandler, Nov 06 2008

A253385 Numbers divisible by at least three distinct primes whose largest prime power factor is not based on its smallest nor its greatest prime factor.

Original entry on oeis.org

90, 126, 180, 252, 270, 350, 360, 378, 504, 525, 540, 550, 594, 630, 650, 700, 702, 756, 810, 825, 850, 918, 950, 975, 1026, 1050, 1078, 1080, 1100, 1134, 1150, 1188, 1242, 1260, 1274, 1275, 1300, 1350, 1400, 1404, 1425, 1512, 1575, 1617, 1620, 1650, 1666, 1700, 1725, 1750, 1782, 1836, 1862, 1890, 1900, 1911, 1950

Offset: 1

Olivier Gérard, Dec 30 2014

This sequence contains all unimodal composites (numbers whose list of prime factors is strictly increasing then strictly decreasing).

			90 is the first member of this sequence because its prime factor decomposition is 2*3^2*5, using the three smallest primes and 3^2 = 9 is the first power of 3 greater than 5 (and 2).

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A057715 (numbers with strictly decreasing prime power factor list).

Module[{pfl},
Select[Range[2000],
  Function[n, pfl = Power @@@ FactorInteger[n];
   1 < First[First[Position[pfl, Max[pfl], 1]]] < Length[pfl]]]]

is(n) = {my(f=factor(n)); if(#f~<3, return(0)); t=max(f[1,1]^f[1,2], f[#f~,1]^f[#f~,2]); for(i=2, #f~, if(f[i, 1] ^ f [i, 2] > t, return(1))) ;0} \\ David A. Corneth, Jun 01 2025

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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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A253385 Numbers divisible by at least three distinct primes whose largest prime power factor is not based on its smallest nor its greatest prime factor.

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