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 1-3 of 3 results.

A085232 In canonical prime factorization: power of smallest prime factor is less than power of greatest prime factor.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 50, 51, 52, 54, 55, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 114, 115
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 22 2003

Keywords

Comments

A028233(a(n)) < A053585(a(n));
p*a(n) is a term for all primes p with A020639(a(n))

A006530(a(n));

a(n)=A057714(n-1) for n<28: a(28)=60, A057714(28-1)=62.

Examples

			60 = 2^2 * 3 * 5 with 2^2=4 < 5, therefore 60 is a term.

Links

Crossrefs

Cf. A085231.

Programs

  • Mathematica
    spfQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]>1&&fi[[1,1]]^fi[[1,2]] < fi[[-1,1]]^fi[[-1,2]]]; Select[Range[120],spfQ] (* Harvey P. Dale, Jul 30 2018 *)

A379094 Numbers whose factors in the canonical prime factorization neither increase weakly nor decrease weakly.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 156, 168, 180, 204, 228, 240, 252, 264, 270, 276, 280, 300, 312, 315, 336, 348, 350, 360, 372, 378, 408, 420, 440, 444, 456, 480, 492, 495, 504, 516, 520, 525, 528, 540, 550, 552, 560, 564, 585, 588, 594, 600, 616, 624, 630, 636, 650
Offset: 1

Views

Author

Peter Luschny, Dec 17 2024

Keywords

Comments

A379097 is a subsequence.
From Michael De Vlieger, Dec 18 2024: (Start)
Proper subset of A126706.
Smallest powerful number is a(314) = 2700. (End)

Examples

			60 is a term because the factors in the canonical prime factorization are [4, 3, 5], a list that is neither increasing nor decreasing.
Primorials (A002110) are not terms of this sequence.

Links

Crossrefs

Cf. A053585, A057714, A085232, A140831, A379097.

Programs

  • Maple
    with(ArrayTools):
fact := n -> local p; [seq(p[1]^p[2], p in ifactors(n)[2])]:
isA379094 := proc(n) local f; f := fact(n);
is(not IsMonotonic(f, direction=decreasing, strict=false) and not IsMonotonic(f, direction=increasing, strict=false)) end:
select(isA379094, [seq(1..650)]);
  • Mathematica
    Select[Range[650], Function[f, NoneTrue[{Sort[f], ReverseSort[f]}, # == f &]][Power @@@ FactorInteger[#]] &] (* Michael De Vlieger, Dec 18 2024 *)
  • PARI
    is_a379094(n) = my(C=apply(x->x[1]^x[2], Vec(factor(n)~))); vecsort(C)!=C && vecsort(C,,4)!=C \\ Hugo Pfoertner, Dec 18 2024

A143907 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

6, 10, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 46, 50, 51, 52, 54, 55, 57, 58, 62, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 114, 115, 116, 117, 118, 119, 122, 123, 124
Offset: 1

Views

Author

Leroy Quet, Sep 04 2008

Keywords

Examples

			2900 is factored as 2^2 * 5^2 * 29^1. Since 2^2 < 5 and 5^2 < 29, then 2900 is in the sequence. On the other hand, 60 is factored as 2^2 * 3^1 * 5^1. Even though 3^1 < 5, 2^2 is not < 3. So 60 is not in the sequence.

Crossrefs

Cf. A057714, A102308.

Programs

  • Mathematica
    okQ[n_] := With[{f = FactorInteger[n]}, If[Length[f] == 1, Return[False]]; For[i = 1, i < Length[f], i++, If[f[[i, 1]]^f[[i, 2]] >= f[[i+1, 1]], Return[False]]]; True]; Select[Range[200], okQ] (* Jean-François Alcover, May 16 2017, adapted from PARI *)
  • PARI
    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

Extensions

Extended by Ray Chandler, Nov 06 2008
Showing 1-3 of 3 results.

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