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.

A325788 Number of complete strict necklace compositions of n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 4, 4, 4, 0, 20, 6, 16, 12, 10, 0, 84, 40, 74, 42, 66, 38, 22, 254, 238, 188, 356, 242, 272, 150, 148, 1140, 1058, 1208, 1546, 1288
Offset: 1

Views

Author

Gus Wiseman, May 22 2019

Keywords

Comments

A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part (counted by A032153). A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is complete if every positive integer from 1 to n is the sum of some circular subsequence.

Examples

			The a(1) = 1 through a(16) = 6 complete strict necklace compositions (empty columns not shown):
  (1)  (12)  (123)  (124)  (1234)  (1253)  (1245)  (1264)  (12345)  (12634)
             (132)  (142)  (1324)  (1325)  (1326)  (1327)  (12354)  (13624)
                           (1423)  (1352)  (1542)  (1462)  (12435)  (14263)
                           (1432)  (1523)  (1623)  (1723)  (12453)  (14326)
                                                           (12543)  (14362)
                                                           (13254)  (16234)
                                                           (13425)
                                                           (13452)
                                                           (13524)
                                                           (13542)
                                                           (14235)
                                                           (14253)
                                                           (14325)
                                                           (14523)
                                                           (14532)
                                                           (15234)
                                                           (15243)
                                                           (15324)
                                                           (15342)
                                                           (15432)

Crossrefs

Cf. A000740, A002033, A008965, A032153, A103295, A126796, A188431, A325684, A325785, A325786, A325787, A325790, A325791.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],neckQ[#]&&Union[Total/@subalt[#]]==Range[n]&]],{n,30}]

A325789 Number of perfect necklace compositions of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 22 2019

Keywords

Comments

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is perfect if every positive integer from 1 to n is the sum of exactly one distinct circular subsequence.

Examples

			The a(1) = 1 , a(2) = 1, a(3) = 2, a(7) = 3, a(13) = 5, and a(31) = 11 perfect necklace compositions (A = 10, B = 11, C = 12, D = 13, E = 14):
  1  11  12   124      1264           12546D
         111  142      1327           1274C5
              1111111  1462           13278A
                       1723           13625E
                       1111111111111  15C472
                                      17324E
                                      1A8723
                                      1D6452
                                      1E4237
                                      1E5263
                                      1111111111111111111111111111111

Crossrefs

Cf. A002033, A008965, A103300, A108917, A126796, A325676, A325679, A325685, A325780, A325782, A325786, A325787, A325789.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&Sort[Total/@subalt[#]]==Range[n]&]],{n,10}]

Formula

For n > 1, a(n) = A325787(n) + 1.

A325990 Numbers with more than one perfect factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456
Offset: 1

Views

Author

Gus Wiseman, May 30 2019

Keywords

Comments

First differs from A060476 in lacking 1 and having 432.
A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

Crossrefs

Positions of terms > 1 in A325989.
Cf. A002033, A292886, A325780, A325781, A325782, A325787, A325789, A325988.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Function[n,Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]]>1]]

A325989 Number of perfect factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 30 2019

Keywords

Comments

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

Examples

			The a(216) = 4 perfect factorizations:
  (2*2*2*3*3*3)
  (2*2*2*3*9)
  (2*3*3*3*4)
  (2*3*4*9)

Crossrefs

Positions of terms > 1 are A325990.
Cf. A002033, A103300, A292886, A325685, A325780, A325787, A325789, A325988.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]],{n,100}]

Formula

a(2^n) = A002033(n).
Showing 1-4 of 4 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.