cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Ralf Stephan, Aug 04 2004

Keywords

Comments

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

Examples

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

Links

Crossrefs

Subset of A024619. Cf. A097319, A097320, A230766.
Cf. A001222, A025487, A118914, A124010, A304678, A329138, A329142.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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]

A071365 Numbers k such that A071364(k) <> A046523(k).

Original entry on oeis.org

18, 50, 54, 75, 90, 98, 108, 126, 147, 150, 162, 198, 234, 242, 245, 250, 270, 294, 300, 306, 324, 338, 342, 350, 363, 375, 378, 414, 450, 486, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 594, 600, 605, 630, 648, 650, 666, 686, 702, 722, 726, 735, 738
Offset: 1

Views

Author

Reinhard Zumkeller, May 21 2002

Keywords

Comments

A071364(k) and A046523(k) have the same prime factors, but not the same sequence of exponents in their prime factorizations.
A046523(k) <> k, as A046523(k) <= A071366(k) <= k.
Numbers with more than one prime factor and, in the ordered factorization, at least one exponent is greater than the previous exponent when read from left to right; contains A097319. - Ray Chandler, Sep 23 2005
Choie et al. call the complementary set of integers (n = p1^e1 * p2^e^2 * ... with exponents e1 >= e2 >= e3 >= ... in their ordered prime factorization) Hardy-Ramanujan integers. - R. J. Mathar, Dec 08 2011
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 6, 71, 759, 7758, 77948, 780216, 7803437, 78033303, 780315757, ... . Apparently, the asymptotic density of this sequence exists and equals 0.07803... . - Amiram Eldar, Aug 04 2024

Examples

			For k = 50 = 2*5*5: A071364(50) = 2*3*3 = 18, A046523(50) = 2*2*3 = 12.
For k = 500 = 2*2*5*5*5: A071364(500) = 2*2*3*3*3 = 108, A046523(500) = 2*2*2*3*3 = 72.

Links

Crossrefs

Cf. A046523, A071364, A071366.

Programs

  • Maple
    a:= proc(n) option remember; local i, k, l;
      for k from 1 +`if`(n=1, 0, a(n-1))
      do l:= sort(ifactors(k)[2], (x, y)->x[1]Alois P. Heinz, Aug 18 2014
  • Mathematica
    Select[Range[750], (e = Last /@ FactorInteger[ # ]) != Sort[e, Greater] &] (* Ray Chandler, Sep 23 2005 *)
Select[Range[750],
OrderedQ[FactorInteger[#][[All, 2]], GreaterEqual] == False &] (* Kenneth A Klinger, Nov 22 2016 *)
  • PARI
    is(k) = {my(e = factor(k)[,2]); e != vecsort(e, , 4);} \\ Amiram Eldar, Aug 04 2024

Extensions

Extended by Ray Chandler, Sep 23 2005

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

Views

Author

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

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

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

Views

Author

Ralf Stephan, Aug 04 2004

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • 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 *)
  • 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

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).

Extensions

Edited by Peter Munn, Jun 01 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

Views

Author

Alex Ratushnyak, Oct 29 2013

Keywords

Links

Crossrefs

Cf. A097318, A097319, A097320.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

If n = Product_{k=1..m} p(k)^e(k), then m > 1 and e(1) <= e(2) <= ... <= e(m).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.