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.

A317090 Positive integers whose prime multiplicities span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85
Offset: 1

Views

Author

Gus Wiseman, Jul 21 2018

Keywords

Comments

The first term in this sequence but absent from A179983 is 180.
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 6, 78, 820, 8379, 84440, 846646, 8473868, 84763404, 847714834, 8477408261, ... . Apparently, the asymptotic density of this sequence exists and equals 0.8477... . - Amiram Eldar, Aug 04 2024

Links

Crossrefs

Cf. A001222, A007916, A055932, A056239, A069799, A112798, A124010, A133808.
Cf. A317087, A317088, A317089, A317091, A317092.
Subsequences: A129912\{1}, A179983\{1}.
Subsequence of A337533.

Programs

  • Mathematica
    normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[2,100],normalQ[FactorInteger[#][[All,2]]]&]
  • PARI
    is(k) = {my(e = Set(factor(k)[,2])); k > 1 && vecmax(e) == #e;} \\ Amiram Eldar, Aug 04 2024

A317089 Numbers whose prime factors span an initial interval of prime numbers and whose prime multiplicities span an initial interval of positive integers.

Original entry on oeis.org

2, 6, 12, 18, 30, 60, 90, 150, 180, 210, 300, 360, 420, 450, 540, 600, 630, 1050, 1260, 1350, 1470, 1500, 2100, 2250, 2310, 2520, 2940, 3150, 3780, 4200, 4410, 4620, 5880, 6300, 6930, 7350, 8820, 9450, 10500, 11550, 12600, 13230, 13860, 14700, 15750, 16170
Offset: 1

Views

Author

Gus Wiseman, Jul 21 2018

Keywords

Examples

			The sequence of rows of A296150 indexed by the terms of this sequence begins: (1), (21), (211), (221), (321), (3211), (3221), (3321), (32211), (4321), (33211), (322111), (43211).

Links

Crossrefs

Cf. A001222, A007916, A055932, A056239, A124010, A133808, A242414, A296150.
Cf. A317087, A317088, A317090, A317091, A317092.

Programs

  • Mathematica
    normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[10000],And[normalQ[PrimePi/@FactorInteger[#][[All,1]]],normalQ[FactorInteger[#][[All,2]]]]&]
  • PARI
    ok(n)={my(f=factor(n), p=f[,1], e=vecsort(f[,2],,8)); n > 1 && #p==primepi(p[#p]) && #e==e[#e]} \\ Andrew Howroyd, Aug 26 2018

A317092 Positive integers whose prime multiplicities are weakly decreasing and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 92
Offset: 1

Views

Author

Gus Wiseman, Jul 21 2018

Keywords

Links

Crossrefs

Cf. A001222, A007916, A055932, A056239, A112798, A124010, A133808, A242031.
Cf. A317087, A317088, A317089, A317090, A317091.

Programs

  • Mathematica
    normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[2,100],And[normalQ[FactorInteger[#][[All,2]]],OrderedQ[Reverse[FactorInteger[#][[All,2]]]]]&]
  • PARI
    is(n) = my (f=factor(n), w=#f~); if (w==0 || f[w,2]!=1, return (0), for (k=1, w-1, if (f[k,2]!=f[k+1,2] && f[k,2]!=1+f[k+1,2], return (0))); return (1)) \\ Rémy Sigrist, Sep 05 2018
Showing 1-3 of 3 results.

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