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 11-19 of 19 results.

A328220 Number of strict integer partitions of n with no pair of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 3, 10, 3, 11, 7, 12, 3, 19, 5, 18, 12, 23, 9, 36, 11, 33, 21, 40, 20, 58, 19, 58, 35, 70, 31, 98, 36, 101, 65, 112, 56, 155, 64, 164, 97, 188, 88, 250, 112, 256, 157, 293, 145, 392, 163, 399, 241, 461, 242
Offset: 0

Views

Author

Gus Wiseman, Oct 14 2019

Keywords

Examples

			The a(2) = 1 through a(20) = 11 partitions (A..K = 10..20):
  2  3  4  5  6   7  8   9   A   B  C    D  E    F   G    H    I    J    K
              42     62  63  64     84      86   96  A6   863  A8   964  C8
                             82     93      A4   A5  C4   962  C6   A63  E6
                                    A2      C2   C3  E2        E4        F5
                                    642     842      862       F3        G4
                                                     A42       G2        I2
                                                               864       A64
                                                               963       A82
                                                               A62       C62
                                                               C42       E42
                                                                         8642

Links

Crossrefs

The non-strict case is A328187.
Partitions with all consecutive parts relatively prime are A328172, with strict case A328188.
Strict partitions with relatively prime parts are A078374.
Partitions with no consecutive divisibilities are A328171.
Cf. A000837, A018783, A070211, A289509, A318980, A325545, A328170, A328221, A328335, A328336.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]==1]&]],{n,0,30}]

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

A328336 Numbers with no consecutive prime indices relatively prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167
Offset: 1

Views

Author

Gus Wiseman, Oct 14 2019

Keywords

Comments

First differs from A318978 in having 897, with prime indices {2, 6, 9}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of partitions no consecutive parts relatively prime (A328187).
Besides the initial 1 this differs from A305078: 47541=897*prime(16) is in A305078 but not in this set. - Andrey Zabolotskiy, Nov 13 2019

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}

Crossrefs

Numbers with consecutive prime indices relatively prime are A328335.
Strict partitions with no consecutive parts relatively prime are A328220.
Numbers with relatively prime prime indices are A289509.
Cf. A000837, A056239, A078374, A112798, A281116, A289508, A318981, A328168, A328169, A328172, A328187, A328188.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MatchQ[primeMS[#],{_,x_,y_,_}/;GCD[x,y]==1]&]

A328188 Number of strict integer partitions of n with all pairs of consecutive parts relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9, 12, 15, 15, 19, 23, 25, 30, 35, 39, 47, 52, 58, 65, 75, 86, 95, 109, 124, 144, 165, 181, 203, 221, 249, 285, 316, 352, 392, 438, 484, 538, 599, 666, 737, 813, 899, 992, 1102, 1215, 1335, 1472, 1621, 1776, 1946, 2137, 2336
Offset: 0

Views

Author

Gus Wiseman, Oct 13 2019

Keywords

Examples

			The a(1) = 1 through a(15) = 15 partitions (A..F = 10..15):
  1  2  3   4   5   6    7   8    9    A     B     C    D     E     F
        21  31  32  51   43  53   54   73    65    75   76    95    87
                41  321  52  71   72   91    74    B1   85    B3    B4
                         61  431  81   532   83    543  94    D1    D2
                             521  432  541   92    651  A3    653   E1
                                  531  721   A1    732  B2    743   654
                                       4321  731   741  C1    752   753
                                             5321  831  652   761   852
                                                   921  751   851   951
                                                        832   941   A32
                                                        5431  A31   B31
                                                        7321  B21   6531
                                                              5432  7431
                                                              6521  7521
                                                              8321  54321

Links

Crossrefs

The case of compositions is A167606.
The non-strict case is A328172.
The Heinz numbers of these partitions are given by A328335.
Partitions with no pairs of consecutive parts relatively prime are A328187.
Cf. A000837, A018783, A178470, A328028, A328170, A328171, A328187, A328220.

Programs

  • Maple
    b:= proc(n, i, s) option remember; `if`(i*(i+1)/2 igcd(i, j)=1, s), b(n-i, min(n-i, i-1),
           numtheory[factorset](i)), 0)+b(n, i-1, s)))
    end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 13 2019
  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{_,x_,y_,_}/;GCD[x,y]>1]&]],{n,0,30}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[i(i + 1)/2 < n, 0, If[n == 0, 1, If[AllTrue[s,  GCD[i, #] == 1&], b[n - i, Min[n - i, i - 1], FactorInteger[i][[All, 1]]], 0] + b[n, i - 1, s]]];
a[n_] := b[n, n, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A328593 Numbers whose binary indices have no consecutive divisible parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 14, 16, 18, 20, 22, 24, 28, 30, 32, 40, 44, 46, 48, 50, 52, 54, 56, 60, 62, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 104, 108, 110, 112, 114, 116, 118, 120, 124, 126, 128, 132, 134, 144, 146, 148, 150, 152, 156, 158, 160
Offset: 1

Views

Author

Gus Wiseman, Oct 21 2019

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Examples

			The sequence of terms together with their binary expansions and binary indices begins:
   0:      0 ~ {}
   1:      1 ~ {1}
   2:     10 ~ {2}
   4:    100 ~ {3}
   6:    110 ~ {2,3}
   8:   1000 ~ {4}
  12:   1100 ~ {3,4}
  14:   1110 ~ {2,3,4}
  16:  10000 ~ {5}
  18:  10010 ~ {2,5}
  20:  10100 ~ {3,5}
  22:  10110 ~ {2,3,5}
  24:  11000 ~ {4,5}
  28:  11100 ~ {3,4,5}
  30:  11110 ~ {2,3,4,5}
  32: 100000 ~ {6}
  40: 101000 ~ {4,6}
  44: 101100 ~ {3,4,6}
  46: 101110 ~ {2,3,4,6}
  48: 110000 ~ {5,6}
  50: 110010 ~ {2,5,6}

Crossrefs

The version for prime indices is A328603.
Numbers with no successive binary indices are A003714.
Partitions with no consecutive divisible parts are A328171.
Compositions without consecutive divisible parts are A328460.
Cf. A000120, A014081, A328187, A328508, A328594, A328595, A328598, A328600, A328608.

Programs

  • Mathematica
    Select[Range[0,100],!MatchQ[Join@@Position[Reverse[IntegerDigits[#,2]],1],{_,x_,y_,_}/;Divisible[y,x]]&]

A328221 Number of integer partitions of n with at least one pair of consecutive divisible parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 10, 12, 20, 26, 38, 51, 73, 92, 126, 166, 219, 283, 369, 470, 604, 763, 968, 1217, 1534, 1907, 2376, 2944, 3640, 4476, 5501, 6723, 8212, 9986, 12130, 14682, 17748, 21376, 25717, 30847, 36959, 44152, 52688, 62714, 74557, 88440, 104775, 123878
Offset: 0

Views

Author

Gus Wiseman, Oct 15 2019

Keywords

Comments

Includes all non-strict partitions.

Examples

			The a(2) = 1 through a(8) = 20 partitions:
  (11)  (21)   (22)    (41)     (33)      (61)       (44)
        (111)  (31)    (221)    (42)      (322)      (62)
               (211)   (311)    (51)      (331)      (71)
               (1111)  (2111)   (222)     (421)      (332)
                       (11111)  (321)     (511)      (422)
                                (411)     (2221)     (431)
                                (2211)    (3211)     (521)
                                (3111)    (4111)     (611)
                                (21111)   (22111)    (2222)
                                (111111)  (31111)    (3221)
                                          (211111)   (3311)
                                          (1111111)  (4211)
                                                     (5111)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

Crossrefs

The complement is counted by A328171.
Partitions whose consecutive parts are relatively prime are A328172.
Partitions with no pair of consecutive parts relatively prime are A328187.
Numbers without consecutive divisible proper divisors are A328028.
Cf. A000837, A018783, A328026, A328161, A328188, A328189, A328194, A328220.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],MatchQ[#,{_,x_,y_,_}/;Divisible[x,y]]&]],{n,0,30}]

A328602 Number of necklace compositions of n where no pair of circularly adjacent parts is relatively prime.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 16, 1, 20, 9, 35, 2, 69, 3, 111, 24, 190, 13, 384, 31, 646, 102, 1212, 113, 2348, 227, 4254, 613, 7993, 976, 15459, 1915, 28825, 4357, 54988, 7868, 105826, 15760, 201115, 33376, 385590, 63974, 744446, 128224, 1428047, 262914, 2754037
Offset: 1

Views

Author

Gus Wiseman, Oct 25 2019

Keywords

Comments

A necklace composition of n (A008965) is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.
Circularity means the last part is followed by the first.

Examples

			The a(2) = 1 through a(10) = 8 necklace compositions:
  (2)  (3)  (4)    (5)  (6)      (7)  (8)        (9)      (10)
            (2,2)       (2,4)         (2,6)      (3,6)    (2,8)
                        (3,3)         (4,4)      (3,3,3)  (4,6)
                        (2,2,2)       (2,2,4)             (5,5)
                                      (2,2,2,2)           (2,2,6)
                                                          (2,4,4)
                                                          (2,2,2,4)
                                                          (2,2,2,2,2)
The a(19) = 3 necklace compositions are: (19), (3,6,4,6), (2,2,6,3,6).

Links

Crossrefs

The non-necklace, non-circular version is A178470.
The version for indivisibility (rather than co-primality) is A328600.
The circularly coprime (as opposed to anti-coprime) version is A328597.
Partitions with no consecutive parts relatively prime are A328187.
Cf. A000031, A000740, A008965, A032153, A318728, A318729, A318748, A328172, A328188, A328220, A328335, A328336, A328601, A328609.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&And@@Not/@CoprimeQ@@@Partition[#,2,1,1]&]],{n,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)={my(v=sum(k=1, n, k*b(n, k, (i,j)->gcd(i,j)<>1))); vector(n, n, sumdiv(n, d, eulerphi(d)*v[n/d])/n)} \\ Andrew Howroyd, Oct 26 2019

Extensions

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

A356736 Heinz numbers of integer partitions with no neighborless parts.

Original entry on oeis.org

1, 6, 12, 15, 18, 24, 30, 35, 36, 45, 48, 54, 60, 72, 75, 77, 90, 96, 105, 108, 120, 135, 143, 144, 150, 162, 175, 180, 192, 210, 216, 221, 225, 240, 245, 270, 288, 300, 315, 323, 324, 360, 375, 384, 385, 405, 420, 432, 437, 450, 462, 480, 486, 525, 539, 540
Offset: 1

Views

Author

Gus Wiseman, Aug 31 2022

Keywords

Comments

First differs from A066312 in having 1 and lacking 462.
First differs from A104210 in having 1 and lacking 42.
A part x is neighborless iff neither x - 1 nor x + 1 are parts.
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.

Examples

			The terms together with their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  30: {1,2,3}
  35: {3,4}
  36: {1,1,2,2}
  45: {2,2,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  60: {1,1,2,3}
  72: {1,1,1,2,2}
  75: {2,3,3}
  77: {4,5}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Crossrefs

These partitions are counted by A355394.
The singleton case is the complement of A356237.
The singleton case is counted by A355393, complement A356235.
The strict complement is A356606, counted by A356607.
The complement is A356734, counted by A356236.
A000041 counts integer partitions, strict A000009.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices of n.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
Cf. A066312, A286470, A287170 (firsts A066205), A328171, A328187, A328221, A328335, A356231, A356234.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Function[ptn,!Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A328675 Number of integer partitions of n with no two distinct consecutive parts divisible.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 13, 13, 22, 23, 30, 36, 50, 54, 77, 85, 113, 135, 170, 194, 256, 303, 369, 440, 545, 640, 792, 931, 1132, 1347, 1616, 1909, 2295, 2712, 3225, 3799, 4519, 5310, 6278, 7365, 8675, 10170, 11928, 13940, 16314, 19046, 22223, 25856
Offset: 0

Views

Author

Gus Wiseman, Oct 29 2019

Keywords

Examples

			The a(1) = 1 through a(10) = 9 partitions (A = 10).
  1  2   3    4     5      6       7        8         9          A
     11  111  22    32     33      43       44        54         55
              1111  11111  222     52       53        72         64
                           111111  322      332       333        73
                                   1111111  2222      432        433
                                            11111111  522        532
                                                      3222       3322
                                                      111111111  22222
                                                                 1111111111

Crossrefs

The Heinz numbers of these partitions are given by A328674.
The case involving all consecutive parts (not just distinct) is A328171.
The version for relative primality instead of divisibility is A328187.
Partitions with all consecutive parts divisible are A003238.
Compositions without consecutive divisibilities are A328460.
Cf. A305148, A316476, A318726, A328172, A328508, A328593, A328598, A328603.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MatchQ[Union[#],{_,x_,y_,_}/;Divisible[y,x]]&]],{n,0,30}]
Previous Showing 11-19 of 19 results.

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