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.

Previous Showing 11-14 of 14 results.

A357485 Heinz numbers of integer partitions with the same length as reverse-alternating sum.

Original entry on oeis.org

1, 2, 20, 42, 45, 105, 110, 125, 176, 182, 231, 245, 312, 374, 396, 429, 494, 605, 663, 680, 702, 780, 782, 845, 891, 969, 1064, 1088, 1100, 1102, 1311, 1426, 1428, 1445, 1530, 1755, 1805, 1820, 1824, 1950, 2001, 2024, 2146, 2156, 2394, 2448, 2475, 2508, 2542
Offset: 1

Views

Author

Gus Wiseman, Oct 01 2022

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i.

Examples

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    20: {1,1,3}
    42: {1,2,4}
    45: {2,2,3}
   105: {2,3,4}
   110: {1,3,5}
   125: {3,3,3}
   176: {1,1,1,1,5}
   182: {1,4,6}
   231: {2,4,5}
   245: {3,4,4}
   312: {1,1,1,2,6}
   374: {1,5,7}
   396: {1,1,2,2,5}

Crossrefs

The version for compositions is A357184, counted by A357182.
These partitions are counted by A357189.
For absolute value we have A357486, counted by A357487.
A000041 counts partitions, strict A000009.
A000712 up to 0's counts partitions w sum = twice alt sum, ranked A349159.
A001055 counts partitions with product equal to sum, ranked by A301987.
A006330 up to 0's counts partitions w sum = twice rev-alt sum, rank A349160.
Cf. A004526, A025047, A051159, A131044, A262046, A357136.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],PrimeOmega[#]==ats[primeMS[#]]&]

A335464 Number of compositions of n with a run of length > 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 8, 18, 39, 86, 188, 406, 865, 1836, 3874, 8135, 17003, 35413, 73516, 152171, 314151, 647051, 1329936, 2728341, 5587493, 11424941, 23327502, 47567628, 96879029, 197090007, 400546603, 813258276, 1649761070, 3343936929, 6772740076, 13707639491
Offset: 0

Views

Author

Gus Wiseman, Jul 06 2020

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n.
Also compositions contiguously matching the pattern (1,1,1).

Examples

			The a(3) = 1 through a(7) = 18 compositions:
  (111)  (1111)  (1112)   (222)     (1114)
                 (2111)   (1113)    (1222)
                 (11111)  (3111)    (2221)
                          (11112)   (4111)
                          (11121)   (11113)
                          (12111)   (11122)
                          (21111)   (11131)
                          (111111)  (13111)
                                    (21112)
                                    (22111)
                                    (31111)
                                    (111112)
                                    (111121)
                                    (111211)
                                    (112111)
                                    (121111)
                                    (211111)
                                    (1111111)

Crossrefs

Compositions contiguously avoiding (1,1) are A003242.
Compositions with some part > 2 are A008466.
Compositions by number of adjacent equal parts are A106356.
Compositions where each part is adjacent to an equal part are A114901.
Compositions contiguously avoiding (1,1,1) are A128695.
Compositions with adjacent parts coprime are A167606.
Compositions contiguously matching (1,1) are A261983.
Compositions with all equal parts contiguous are A274174.
Patterns contiguously matched by compositions are A335457.
Cf. A005251, A011782, A056986, A032020, A131044, A178470, A242882, A335455, A335458.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, add(`if`(abs(t)<>j,
       b(n-j, j), `if`(t=-j, 0, b(n-j, -j))), j=1..n))
    end:
a:= n-> ceil(2^(n-1))-b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 06 2020
  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,x_,x_,_}]&]],{n,0,10}]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[If[Abs[t] != j,
     b[n - j, j], If[t == -j, 0, b[n - j, -j]]], {j, 1, n}]];
a[n_] := Ceiling[2^(n-1)] - b[n, 0];
a /@ Range[0, 40] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

Formula

a(n) = A011782(n) - A128695(n). - Alois P. Heinz, Jul 06 2020

Extensions

a(23)-a(35) from Alois P. Heinz, Jul 06 2020

A357487 Number of integer partitions of n with the same length as reverse-alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 2, 0, 4, 0, 5, 0, 9, 0, 13, 0, 23, 0, 34, 0, 54, 0, 78, 0, 120, 0, 170, 0, 252, 0, 358, 0, 517, 0, 725, 0, 1030, 0, 1427, 0, 1992, 0, 2733, 0, 3759, 0, 5106, 0, 6946, 0, 9345, 0, 12577, 0, 16788, 0, 22384, 0, 29641, 0
Offset: 0

Views

Author

Gus Wiseman, Oct 01 2022

Keywords

Comments

A partition of n is a weakly decreasing sequence of positive integers summing to n.
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i.

Examples

			The a(1) = 1 through a(13) = 9 partitions:
  1   .  .  .  311   .  322   .  333     .  443     .  553
                        421      432        542        652
                                 531        641        751
                                 51111      52211      52222
                                            62111      53311
                                                       62221
                                                       63211
                                                       73111
                                                       7111111

Crossrefs

For product equal to sum we have A001055, compositions A335405.
The version for compositions is A357182, reverse ranked by A357184.
The reverse version is A357189, ranked by A357486.
These partitions are ranked by A357485.
Removing zeros gives A357488.
A000041 counts partitions, strict A000009.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
Cf. A004526, A051159, A114220, A131044, A262046, A262977, A301987, A357183.

Programs

  • Mathematica
    ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[Reverse[#]]&]],{n,0,30}]

A357486 Heinz numbers of integer partitions with the same length as alternating sum.

Original entry on oeis.org

1, 2, 10, 20, 21, 42, 45, 55, 88, 91, 105, 110, 125, 156, 176, 182, 187, 198, 231, 245, 247, 312, 340, 351, 374, 390, 391, 396, 429, 494, 532, 544, 550, 551, 605, 663, 680, 702, 713, 714, 765, 780, 782, 845, 891, 910, 912, 969, 975, 1012, 1064, 1073, 1078
Offset: 1

Views

Author

Gus Wiseman, Oct 01 2022

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    10: {1,3}
    20: {1,1,3}
    21: {2,4}
    42: {1,2,4}
    45: {2,2,3}
    55: {3,5}
    88: {1,1,1,5}
    91: {4,6}
   105: {2,3,4}
   110: {1,3,5}
   125: {3,3,3}
   156: {1,1,2,6}
   176: {1,1,1,1,5}

Crossrefs

For product instead of length we have new, counted by A004526.
The version for compositions is A357184, counted by A357182.
For absolute value we have A357486, counted by A357487.
These partitions are counted by A357189.
A000041 counts partitions, strict A000009.
A000712 up to 0's counts partitions, sum = twice alt sum, rank A349159.
A001055 counts partitions with product equal to sum, ranked by A301987.
A006330 up to 0's counts partitions, sum = twice rev-alt sum, rank A349160.
A025047 counts alternating compositions.
A357136 counts compositions by alternating sum.
Cf. A051159, A131044, A262046.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],PrimeOmega[#]==ats[Reverse[primeMS[#]]]&]
Previous Showing 11-14 of 14 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.