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.

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A318726 Number of integer compositions of n that have only one part or whose consecutive parts are indivisible and the last and first part are also indivisible.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 3, 8, 13, 12, 23, 27, 56, 64, 100, 150, 216, 325, 459, 700, 1007, 1493, 2186, 3203, 4735, 6929, 10243, 14952, 22024, 32366, 47558, 69906, 102634, 150984, 221713, 325919, 478842, 703648, 1034104, 1519432, 2233062, 3281004, 4821791, 7085359
Offset: 1

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Examples

			The a(10) = 13 compositions:
  (10)
  (7,3) (3,7) (6,4) (4,6)
  (5,3,2) (5,2,3) (3,5,2) (3,2,5) (2,5,3) (2,3,5)
  (3,2,3,2) (2,3,2,3)
The a(11) = 12 compositions:
  (11)
  (9,2) (2,9) (8,3) (3,8) (7,4) (4,7) (6,5) (5,6)
  (5,2,4) (4,5,2) (2,4,5)

Links

Crossrefs

Cf. A000740, A008965, A167606, A285573, A296302, A303362, A304713, A316476, A318727.

Programs

  • Mathematica
    Table[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,({_,x_,y_,_}/;Divisible[x,y])|({y_,_,x_}/;Divisible[x,y])]&]//Length,{n,20}]
  • PARI
    b(n,k,pred)={my(M=matrix(n,n)); for(n=1, n, M[n,n]=pred(k,n); for(j=1, n-1, M[n,j]=sum(i=1, n-j, if(pred(i,j), M[n-j,i], 0)))); sum(i=1, n, if(pred(i,k), M[n,i], 0))}
a(n)={1 + sum(k=1, n-1, b(n-k, k, (i,j)->i%j<>0))} \\ Andrew Howroyd, Sep 08 2018

Formula

a(n) = A328598(n) + 1. - Gus Wiseman, Nov 04 2019

Extensions

a(21)-a(28) from Robert Price, Sep 08 2018
Terms a(29) and beyond from Andrew Howroyd, Sep 08 2018
Name corrected by Gus Wiseman, Nov 04 2019

A318728 Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose adjacent parts (including the last with first) are coprime.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 22, 34, 58, 95, 168, 280, 492, 853, 1508, 2648, 4715, 8350, 14924, 26643, 47794, 85779, 154475, 278323, 502716, 908913, 1646206, 2984547, 5418653, 9847190, 17916001, 32625618, 59470540, 108493150, 198094483, 361965239, 661891580, 1211162271
Offset: 1

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Examples

			The a(7) = 13 cyclic compositions with adjacent parts coprime:
  7,
  16, 25, 34,
  115,
  1114, 1213, 1132, 1123,
  11113, 11212,
  111112,
  1111111.

Links

Crossrefs

Cf. A000740, A000837, A008965, A059966, A100953, A167606, A296302, A318726, A318727, A328597.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,neckQ[#]&&And@@CoprimeQ@@@Partition[#,2,1,1]]&]],{n,20}]
  • PARI
    b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->gcd(i,j)==1))); vector(n, n, (n > 1) + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019

Formula

a(n) = A328597(n) + 1 for n > 1. - Andrew Howroyd, Oct 27 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018
Name corrected by Gus Wiseman, Nov 04 2019

A318729 Number of cyclic compositions (necklaces of positive integers) summing to n that have only one part or whose consecutive parts (including the last with first) are indivisible.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 4, 6, 6, 8, 11, 19, 21, 30, 41, 59, 79, 112, 157, 219, 305, 430, 605, 860, 1210, 1727, 2424, 3463, 4905, 7001, 9954, 14211, 20271, 28980, 41392, 59254, 84800, 121540, 174163, 249932, 358578, 515091, 739933, 1063827, 1529767, 2201383
Offset: 1

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Examples

			The a(13) = 11 cyclic compositions with successive parts indivisible:
  (13)
  (2,11) (3,10) (4,9) (5,8) (6,7)
  (2,4,7) (2,6,5) (2,8,3) (3,6,4)
  (2,3,5,3)

Links

Crossrefs

Cf. A000740, A008965, A059966, A167606, A285573, A303362, A304713, A316476, A318726, A318727, A318728, A318730, A328600.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,neckQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]]&]],{n,20}]
  • PARI
    b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i,j)->i%j<>0))); vector(n, n, 1 + sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 27 2019

Formula

a(n) = A328600(n) + 1. - Andrew Howroyd, Oct 27 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018
Name corrected by Gus Wiseman, Nov 04 2019

A342334 Number of compositions of n with all adjacent parts (x, y) satisfying x >= 2y or y > 2x.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 11, 16, 23, 35, 54, 82, 125, 193, 294, 447, 680, 1037, 1580, 2408, 3676, 5606, 8544, 13024, 19860, 30277, 46155, 70374, 107300, 163586, 249397, 380235, 579705, 883810, 1347467, 2054371, 3132102, 4775211, 7280321, 11099613, 16922503, 25800136, 39335052, 59970425, 91431195
Offset: 0

Views

Author

Gus Wiseman, Mar 10 2021

Keywords

Comments

Also the number of compositions of n with all adjacent parts (x, y) satisfying x > 2y or y >= 2x.

Examples

			The a(1) = 1 through a(8) = 16 compositions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (12)  (13)  (14)   (15)   (16)    (17)
                  (31)  (41)   (24)   (25)    (26)
                        (131)  (51)   (52)    (62)
                               (141)  (61)    (71)
                               (312)  (124)   (125)
                                      (151)   (152)
                                      (241)   (161)
                                      (313)   (251)
                                      (412)   (314)
                                      (1312)  (413)
                                              (512)
                                              (1241)
                                              (1313)
                                              (1412)
                                              (3131)

Crossrefs

The unordered version (partitions) is A342098 or A000929 (multisets).
The version not allowing equality (i.e., strict relations) is A342332.
The version allowing equality (i.e., non-strict relations) is A342333.
Reversing operators and changing 'or' into 'and' gives A342338.
A002843 counts compositions with adjacent parts x <= 2y.
A154402 counts partitions with adjacent parts x = 2y.
A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent parts x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342337 counts partitions with adjacent parts x = y or x = 2y.
Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342336, A342340.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]>=2*#[[i-1]]||#[[i-1]]>2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]

Extensions

More terms from Joerg Arndt, Mar 12 2021

A342342 Number of strict compositions of n with all adjacent parts (x, y) satisfying x <= 2y and y <= 2x.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 5, 5, 3, 11, 9, 11, 17, 15, 29, 39, 31, 39, 65, 57, 107, 127, 149, 155, 187, 265, 293, 419, 523, 571, 781, 763, 941, 1371, 1387, 2125, 2383, 2775, 3243, 4189, 4555, 5349, 7241, 7997, 10591, 13171, 14581, 17213, 20253, 25177, 27701, 34317
Offset: 0

Views

Author

Gus Wiseman, Mar 12 2021

Keywords

Comments

Each quotient of adjacent parts is between 1/2 and 2 inclusive.

Examples

			The a(1) = 1 through a(12) = 17 strict compositions (A = 10, B = 11, C = 12):
  1   2   3    4   5    6     7     8    9     A      B      C
          12       23   24    34    35   36    46     47     48
          21       32   42    43    53   45    64     56     57
                        123   124        54    235    65     75
                        321   421        63    532    74     84
                                         234   1234   236    246
                                         243   1243   245    345
                                         324   3421   542    354
                                         342   4321   632    435
                                         423          1235   453
                                         432          5321   534
                                                             543
                                                             642
                                                             1236
                                                             1245
                                                             5421
                                                             6321

Crossrefs

The non-strict version is A224957.
The case with strict relations is A342341 (non-strict: A342330).
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A154402 counts partitions with adjacent parts x = 2y.
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342337 counts partitions with adjacent parts x = y or x = 2y.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342334, A342336, A342340.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],And@@Table[#[[i]]<=2*#[[i-1]]&&#[[i-1]]<=2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]

Extensions

a(40)-a(51) from Alois P. Heinz, May 24 2021

A328598 Number of compositions of n with no part circularly followed by a divisor.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 0, 4, 2, 7, 12, 11, 22, 26, 55, 63, 99, 149, 215, 324, 458, 699, 1006, 1492, 2185, 3202, 4734, 6928, 10242, 14951, 22023, 32365, 47557, 69905, 102633, 150983, 221712, 325918, 478841, 703647, 1034103, 1519431, 2233061, 3281003, 4821790, 7085358
Offset: 0

Views

Author

Gus Wiseman, Oct 24 2019

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n.
Circularity means the last part is followed by the first.

Examples

			The a(5) = 2 through a(12) = 22 compositions (empty column not shown):
  (2,3)  (2,5)  (3,5)  (2,7)    (3,7)      (2,9)    (5,7)
  (3,2)  (3,4)  (5,3)  (4,5)    (4,6)      (3,8)    (7,5)
         (4,3)         (5,4)    (6,4)      (4,7)    (2,3,7)
         (5,2)         (7,2)    (7,3)      (5,6)    (2,7,3)
                       (2,4,3)  (2,3,5)    (6,5)    (3,2,7)
                       (3,2,4)  (2,5,3)    (7,4)    (3,4,5)
                       (4,3,2)  (3,2,5)    (8,3)    (3,5,4)
                                (3,5,2)    (9,2)    (3,7,2)
                                (5,2,3)    (2,4,5)  (4,3,5)
                                (5,3,2)    (4,5,2)  (4,5,3)
                                (2,3,2,3)  (5,2,4)  (5,3,4)
                                (3,2,3,2)           (5,4,3)
                                                    (7,2,3)
                                                    (7,3,2)
                                                    (2,3,2,5)
                                                    (2,3,4,3)
                                                    (2,5,2,3)
                                                    (3,2,3,4)
                                                    (3,2,5,2)
                                                    (3,4,3,2)
                                                    (4,3,2,3)
                                                    (5,2,3,2)

Links

Crossrefs

The necklace version is A328600, or A318729 without singletons.
The version with singletons is A318726.
The non-circular version is A328460.
Also forbidding parts circularly followed by a multiple gives A328599.
Partitions with no part followed by a divisor are A328171.
Cf. A000740, A008965, A167606, A178470, A318748, A328187, A328508, A328593, A328597, A328601, A328603, A328609.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Not/@Divisible@@@Partition[#,2,1,1]&]],{n,0,10}]
  • PARI
    b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q,]}
seq(n)={concat([1], sum(k=1, n, b(n, k, (i,j)->i%j<>0)))} \\ Andrew Howroyd, Oct 26 2019

Formula

a(n > 0) = A318726(n) - 1.

Extensions

Terms a(26) and beyond from Andrew Howroyd, Oct 26 2019

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

A318748 Number of integer compositions of n that have only one part or whose consecutive parts are coprime and the last and first part are also coprime.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 43, 82, 151, 285, 535, 1005, 1883, 3533, 6631, 12460, 23407, 43952, 82538, 154999, 291088, 546674, 1026687, 1928118, 3621017, 6800300, 12771086, 23984329, 45042959, 84591339, 158863807, 298348613, 560303342, 1052258402, 1976157510
Offset: 0

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Examples

			The a(5) = 13 compositions with adjacent parts coprime:
  (5)
  (41) (14) (32) (23)
  (311) (131) (113)
  (2111) (1211) (1121) (1112)
  (11111)
Missing from this list are (221), (212), and (122).

Links

Crossrefs

Cf. A000740, A008965, A059966, A100953, A167606, A296302, A318726, A318727, A318728, A318745, A328609.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,And@@CoprimeQ@@@Partition[#,2,1,1]]&]],{n,20}]
  • PARI
    b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={concat([1], vector(n, i, i > 1) + sum(k=1, n, b(n, k, (i, j)->gcd(i, j)==1)))} \\ Andrew Howroyd, Nov 01 2019

Formula

a(n) = A328609(n) + 1 for n > 1. - Andrew Howroyd, Nov 01 2019

Extensions

a(21)-a(35) from Alois P. Heinz, Sep 02 2018
Name corrected by Gus Wiseman, Nov 04 2019

A342495 Number of compositions of n with constant (equal) first quotients.

Original entry on oeis.org

1, 1, 2, 4, 5, 6, 8, 10, 10, 11, 12, 12, 16, 16, 18, 20, 19, 18, 22, 22, 24, 28, 24, 24, 30, 27, 30, 30, 34, 30, 38, 36, 36, 36, 36, 40, 43, 40, 42, 46, 48, 42, 52, 46, 48, 52, 48, 48, 56, 55, 54, 54, 58, 54, 60, 58, 64, 64, 60, 60, 72, 64, 68, 74, 69, 72, 72
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2021

Keywords

Comments

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The composition (1,2,4,8) has first quotients (2,2,2) so is counted under a(15).
The composition (4,5,6) has first quotients (5/4,6/5) so is not counted under a(15).
The a(1) = 1 through a(7) = 10 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (12)   (13)    (14)     (15)      (16)
             (21)   (22)    (23)     (24)      (25)
             (111)  (31)    (32)     (33)      (34)
                    (1111)  (41)     (42)      (43)
                            (11111)  (51)      (52)
                                     (222)     (61)
                                     (111111)  (124)
                                               (421)
                                               (1111111)

Links

Crossrefs

The version for differences instead of quotients is A175342.
The unordered version is A342496, ranked by A342522.
The strict unordered version is A342515.
The distinct version is A342529.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A002843, A003242, A008965, A048004, A059966, A074206, A167606, A253249, A318991, A318992, A325557, A342528.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]

Formula

a(n > 0) = 2*A342496(n) - A000005(n).

A342529 Number of compositions of n with distinct first quotients.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 19, 36, 67, 114, 197, 322, 564, 976, 1614, 2729, 4444, 7364, 12357, 20231, 33147, 53973, 87254, 140861, 227535, 368050, 589706, 940999, 1497912, 2378260, 3774297, 5964712, 9416411, 14822087, 23244440, 36420756
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2021

Keywords

Comments

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The composition (2,1,2,3) has first quotients (1/2,2,3/2) so is counted under a(8).
The a(1) = 1 through a(5) = 13 compositions:
  (1)  (2)    (3)    (4)      (5)
       (1,1)  (1,2)  (1,3)    (1,4)
              (2,1)  (2,2)    (2,3)
                     (3,1)    (3,2)
                     (1,1,2)  (4,1)
                     (1,2,1)  (1,1,3)
                     (2,1,1)  (1,2,2)
                              (1,3,1)
                              (2,1,2)
                              (2,2,1)
                              (3,1,1)
                              (1,1,2,1)
                              (1,2,1,1)

Links

Crossrefs

The version for differences instead of quotients is A325545.
The version for equal first quotients is A342495.
The unordered version is A342514, ranked by A342521.
The strict unordered version is A342520.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A003242, A008965, A048004, A059966, A167606, A253249, A318991, A318992, A342527, A342528.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]

Extensions

a(21)-a(35) from Alois P. Heinz, Jan 16 2025
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