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 1-4 of 4 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

A375402 Numbers whose maximal anti-runs of weakly increasing prime factors (with multiplicity) have distinct maxima.

Original entry on oeis.org

1, 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
Offset: 1

Views

Author

Gus Wiseman, Aug 14 2024

Keywords

Comments

First differs from A349810 in lacking 150.
An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.
The partitions with these Heinz numbers are those with (1) no part appearing more than twice and (2) the greatest part appearing only once.
Note the prime factors can alternatively be written in weakly decreasing order.
How is does the sequence relate to A317092? - R. J. Mathar, Aug 20 2024

Examples

			The maximal anti-runs of prime factors of 150 are ((2,3,5),(5)), with maxima (5,5), so 150 is not in the sequence.
The maximal anti-runs of prime factors of 180 are ((2),(2,3),(3,5)), with maxima (2,3,5), so 180 is in the sequence.
The maximal anti-runs of prime factors of 300 are ((2),(2,3,5),(5)), with maxima (2,5,5), so 300 is not in the sequence.

Crossrefs

For identical instead of distinct we have A065200, complement A065201.
A version for compositions (instead of partitions) is A374767.
Partitions of this type are counted by A375133.
For minima instead of maxima we have A375398, counted by A375134.
The complement for minima is A375399, counted by A375404.
The complement is A375403, counted by A375401.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A046660, A066328, A358830, A374632, A374706, A374768, A375128, A375136, A375396, A375400.

Programs

  • Mathematica
    Select[Range[150],UnsameQ@@Max /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

A317091 Positive integers whose prime multiplicities are weakly increasing and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 101, 102
Offset: 1

Views

Author

Gus Wiseman, Jul 21 2018

Keywords

Links

Crossrefs

Cf. A001222, A007916, A055932, A056239, A112798, A124010, A133808, A304678.
Cf. A317087, A317088, A317089, A317090, A317092.

Programs

  • Mathematica
    normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[2,150],And[normalQ[FactorInteger[#][[All,2]]],OrderedQ[FactorInteger[#][[All,2]]]]&]
Showing 1-4 of 4 results.

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