A097320 -id:A097320 - 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]

A096156 Numbers with ordered prime signature (2,1).

Original entry on oeis.org

12, 20, 28, 44, 45, 52, 63, 68, 76, 92, 99, 116, 117, 124, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 244, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 356, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452, 475, 477, 508, 524, 531, 539, 548, 549

Offset: 1

Alford Arnold, Jul 24 2004

Numbers of the form p^2 * q where p and q are primes with p < q.
Also terms of A054753 that are not in A095990.
There are pairs that differ by 1, which is not the case in A095990, beginning with 44 and 45, 116 and 117, 171 and 172, 332 and 333, etc.

			a(2) = 20 because 20 = 2*2*5 and 2 < 5.
Note that 18 = 2*3^2 is not in the sequence, even though it has prime signature (2,1), because its ordered prime signature is (1,2) (A095990). Prime signatures correspond to partitions of Omega(n), while ordered prime signatures correspond to compositions of Omega(n).

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
OEIS Wiki, Ordered prime signatures

Cf. A095990.

Subsequence of A054753, A097320, A325241, A345381.

Take[ Sort[ Flatten[ Table[ Prime[p]^2 Prime[q], {q, 2, 33}, {p, q - 1}]]], 54] (* Robert G. Wilson v, Jul 28 2004 *)
Select[Range[10^5],FactorInteger[#][[All,2]]=={2,1}&] (* Enrique Pérez Herrero, Jun 27 2012 *)

list(lim)=my(v=List()); forprime(q=3, lim\4, forprime(p=2, min(sqrtint(lim\q), q-1), listput(v, p^2*q))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014

from sympy import factorint
def ok(n): return list(factorint(n).values()) == [2, 1]
print([k for k in range(550) if ok(k)]) # Michael S. Branicky, Dec 20 2021

Edited and extended by Robert G. Wilson v and Rick L. Shepherd, Jul 27 2004

A112769 Numbers with more than one prime factor and, in the ordered factorization, at least one exponent is less than the previous exponent when read from left to right.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212, 220

Offset: 1

Ray Chandler, Sep 23 2005, based on a suggestion from Leroy Quet

nonn

This sequence lists the integers x such that A085079(x) > x. - Michel Marcus, Jun 25 2025 and Jul 30 2015

			90 = 2^1 * 3^2 * 5^1 and 2 > 1, so 90 is in sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A085079, A097318, A097319, A097320, A304678.

mopfQ[n_]:=Module[{e=FactorInteger[n][[All,2]]},Length[e]>1&&Min[ Differences[ e]]<0]; Select[Range[300],mopfQ] (* Harvey P. Dale, May 30 2018 *)

isok(n) = {f = factor(n)[,2]; if (#f > 1, for (k=2, #f, if (f[k] < f[k-1], return (1)););); return (0);} \\ Michel Marcus, Jul 30 2015

A087315 a(n) = Product_{k=1..n} prime(k)^prime(n-k+1).

Original entry on oeis.org

1, 4, 72, 21600, 190512000, 580909190400000, 428616352408083840000000, 859278392084450410309036800000000000, 2097197194438629126172451944256706311040000000000000

Offset: 0

Amarnath Murthy, Sep 03 2003

nonn

			a(3) = 2^5*3^3*5^2 = 21600.

G. C. Greubel, Table of n, a(n) for n = 0..23

Cf. A006939, A002110, A025487, A004394, A002093, A002182, A055932, A025487, A089247, A071364, A097320, A046523, A056166, A114129, A053810.

[1] cat [(&*[NthPrime(k)^(NthPrime(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

seq(product(ithprime(k)^ithprime(n-k+1), k=1..n), n=0..10);

Table[Product[Prime[k]^Prime[n - k + 1], {k, 1, n}], {n, 0, 10}] (* G. C. Greubel, Oct 14 2018 *)

for(n=0, 10, print1(prod(k=1,n, prime(k)^prime(n-k+1)), ", ")) \\ G. C. Greubel, Oct 14 2018

[prod(nth_prime(i)^nth_prime(k-i+1) for i in (1..k)) for k in (0..10)] # Giuseppe Coppoletta, Nov 03 2014

More terms from Jorge Coveiro, Dec 22 2004

Corrected by David Wasserman, May 02 2005

A097319 Numbers with more than one prime factor and, in the ordered factorization, the exponents are strictly increasing.

Original entry on oeis.org

18, 50, 54, 75, 98, 108, 147, 162, 242, 245, 250, 324, 338, 363, 375, 486, 500, 507, 578, 605, 648, 686, 722, 845, 847, 867, 972, 1029, 1058, 1083, 1125, 1183, 1250, 1372, 1445, 1458, 1587, 1682, 1715, 1805, 1859, 1875, 1922, 1944, 2023, 2250

Offset: 1

Ralf Stephan, Aug 04 2004

nonn

If n = Product[k=1..m, p(k)^e(k)], then m>1 and e(1) < e(2) <...< e(m).

			507 is 3^1*13^2, A001221(507)=2 and 1<2, so 507 is in sequence.
150 is 2^1*3^1*5^2 is not in the sequence because 1,1,2 is not strictly increasing (although it is nondecreasing).

T. D. Noe, Table of n, a(n) for n = 1..1180

Subset of A126706. Cf. A097318, A097320.

fQ[n_] := Module[{d, f = FactorInteger[n]}, If[Length[f] == 1, False, d = Differences[Transpose[f][[2]]]; And @@ ((# > 0) & /@ d)]]; Select[Range[2250], fQ] (* T. D. Noe, Apr 09 2013 *)

for(n=1, 3000, 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, , 8) == f \\ Rick L. Shepherd, Jan 17 2018

A057715 Numbers m = Product p_i^{e_i}, not a power of a prime, such that p_j^{e_j} > p_k^{e_k} for all p_j < p_k.

Original entry on oeis.org

12, 24, 40, 45, 48, 56, 63, 80, 96, 112, 135, 144, 160, 175, 176, 189, 192, 208, 224, 275, 288, 297, 320, 325, 351, 352, 384, 405, 416, 425, 448, 459, 475, 513, 539, 544, 567, 575, 576, 608, 621, 637, 640, 675, 704, 720, 736, 768, 800, 832, 833, 864, 875

Offset: 1

Leroy Quet, Oct 24 2000

			720 is included because 720 = 2^4 * 3^2 * 5^1 and 2^4 > 3^2 > 5^1.

Lei Zhou, Table of n, a(n) for n = 1..10000

Subsequence of A085231, A097320, A126706.

Cf. A363063.

Select[Range[575], Greater @@ Power @@@ (fi = FactorInteger[#]) && Length[fi] > 1 &] (* Ray Chandler, Nov 06 2008 *)

Title clarified by Sean A. Irvine, Jun 24 2022 and Peter Munn, May 26 2025

A230766 Numbers with more than one prime factor and, in the ordered factorization, the exponent never decreases when read from left to right.

Original entry on oeis.org

6, 10, 14, 15, 18, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 50, 51, 54, 55, 57, 58, 62, 65, 66, 69, 70, 74, 75, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 98, 100, 102, 105, 106, 108, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141

Offset: 1

Alex Ratushnyak, Oct 29 2013

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A097318, A097319, A097320.

filter:= proc(n) local F;
  F:= sort(ifactors(n)[2],(a,b) -> a[1] < b[1]);
  if nops(F) = 1 then return false fi;
  F:= F[..,2];
  F = sort(F)
end proc:
select(filter, [$2..200]); # Robert Israel, Feb 07 2025

fQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Length[f] > 1 && Min[Differences[f]] >= 0]; Select[Range[2, 200], fQ] (* T. D. Noe, Nov 04 2013 *)
Select[Range[150],PrimeNu[#]>1&&Min[Differences[FactorInteger[#][[All,2]]]]>=0&] (* Harvey P. Dale, May 22 2020 *)

isok(n) = {my(f = factor(n), nbf = #f~); if (nbf < 2, return (0)); lastexp = 0; for (i=1, nbf, if ((newexp = f[i, 2]) < lastexp, return (0)); lastexp = newexp;); return (1);} \\ Michel Marcus, Oct 30 2013

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

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A096156 Numbers with ordered prime signature (2,1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A112769 Numbers with more than one prime factor and, in the ordered factorization, at least one exponent is less than the previous exponent when read from left to right.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A087315 a(n) = Product_{k=1..n} prime(k)^prime(n-k+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Extensions

A097319 Numbers with more than one prime factor and, in the ordered factorization, the exponents are strictly increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A057715 Numbers m = Product p_i^{e_i}, not a power of a prime, such that p_j^{e_j} > p_k^{e_k} for all p_j < p_k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A230766 Numbers with more than one prime factor and, in the ordered factorization, the exponent never decreases when read from left to right.

Views

Author

Keywords

Links

Crossrefs

Programs