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.

A325757 Irregular triangle read by rows giving the frequency span of n.

Original entry on oeis.org

1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 4, 1, 1, 1, 3, 2, 2, 2, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 2, 2, 6, 1, 1, 1, 2, 4, 1, 1, 2, 2, 3, 1, 1, 1, 1, 4, 7, 1, 1, 1, 1, 2, 2, 2, 2, 8, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 4, 1, 1, 1, 2, 5, 9, 1, 1, 1, 1, 1, 1, 2, 2, 3
Offset: 1

Views

Author

Gus Wiseman, May 19 2019

Keywords

Comments

We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer is the frequency span of its prime indices (row n of A296150).

Examples

			Triangle begins:
   1:
   2: 1
   3: 2
   4: 1 1 2
   5: 3
   6: 1 1 1 2 2
   7: 4
   8: 1 1 1 3
   9: 2 2 2
  10: 1 1 1 2 3
  11: 5
  12: 1 1 1 1 1 2 2 2
  13: 6
  14: 1 1 1 2 4
  15: 1 1 2 2 3
  16: 1 1 1 1 4
  17: 7
  18: 1 1 1 1 2 2 2 2
  19: 8
  20: 1 1 1 1 1 2 2 3
  21: 1 1 2 2 4
  22: 1 1 1 2 5
  23: 9
  24: 1 1 1 1 1 1 2 2 3
  25: 2 3 3
  26: 1 1 1 2 6
  27: 2 2 2 3
  28: 1 1 1 1 1 2 2 4

Crossrefs

Row lengths are A325249.
Run-lengths are A325758.
Number of distinct terms in row n is A325759(n).
Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A290822, A323014, A324843, A325277, A325755, A325760.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
Table[freqspan[primeMS[n]],{n,15}]

A325758 Irregular triangle read by rows giving the frequency span signature of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 2, 1, 3, 1, 3, 3, 1, 1, 1, 5, 3, 1, 3, 1, 1, 2, 2, 1, 4, 1, 1, 4, 4, 1, 5, 2, 1, 2, 2, 1, 3, 1, 1, 1, 6, 2, 1, 1, 2, 3, 1, 1, 3, 1, 5, 2, 1, 1, 4, 1, 2, 1, 5, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 5, 1, 3, 1, 1, 2, 2, 1, 6, 1, 2, 1, 4, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 19 2019

Keywords

Comments

We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer is the frequency span of its prime indices (row n of A296150). Row n of this triangle gives an unsorted list of the multiplicities in the frequency span of n. For example, the frequency span of 30 is {1,1,1,1,2,3,3}, so row 30 is (4,1,2).

Examples

			Triangle begins:
  1
  1
  2 1
  1
  3 2
  1
  3 1
  3
  3 1 1
  1
  5 3
  1
  3 1 1
  2 2 1
  4 1
  1
  4 4
  1
  5 2 1
  2 2 1
  3 1 1
  1
  6 2 1
  1 2
  3 1 1
  3 1
  5 2 1
  1
  4 1 2

Crossrefs

Row sums are A325249.
Row lengths are A325759.
Run-lengths of A325757.
Row n is the unsorted prime signature of A325760(n).
Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A290822, A323014,A325277.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
Table[Length/@Split[freqspan[primeMS[n]]],{n,30}]

A325759 Number of distinct frequencies in the frequency span of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 3, 2, 1, 2, 1, 3, 3, 3, 1, 3, 2, 3, 2, 3, 1, 3, 1, 2, 3, 3, 4, 2, 1, 3, 3, 3, 1, 4, 1, 3, 3, 3, 1, 3, 2, 3, 3, 3, 1, 3, 4, 4, 3, 3, 1, 3, 1, 3, 3, 2, 4, 4, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 4, 4, 1, 4, 2, 3, 1, 3, 4, 3, 3
Offset: 1

Views

Author

Gus Wiseman, May 19 2019

Keywords

Comments

We define the frequency span of an integer partition to be the partition itself if it has no or only one block, and otherwise it is the multiset union of the partition and the frequency span of its multiplicities. For example, the frequency span of (3,2,2,1) is {1,2,2,3} U {1,1,2} U {1,2} U {1,1} U {2} = {1,1,1,1,1,1,2,2,2,2,2,3}. The frequency span of a positive integer n is the frequency span of its prime indices (row n of A296150).

Crossrefs

Row lengths of A325758.
Number of distinct entries in row n of A325757.
Cf. A001221, A001222, A056239, A071625, A112798, A181819, A182857, A323014, A325249, A325277, A325760.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
freqspan[ptn_]:=If[Length[ptn]<=1,ptn,Sort[Join[ptn,freqspan[Sort[Length/@Split[ptn]]]]]];
Table[Length[Union[freqspan[primeMS[n]]]],{n,100}]
Showing 1-3 of 3 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.