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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A329143 Number of integer partitions of n whose augmented differences are a periodic word.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 3, 2, 2, 3, 2, 2, 4, 4, 5, 3, 5, 2, 10, 5, 6, 5, 10, 5, 11, 7, 13, 6, 15, 6, 20, 11, 18, 12, 27, 8, 27, 16, 32, 14, 35, 14, 42, 23, 43, 17, 56, 17, 61, 31, 67, 25, 78, 28, 88, 41, 89, 35, 119, 39, 116, 60, 131, 52, 154, 52, 170, 75, 182
Offset: 0

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Author

Gus Wiseman, Nov 10 2019

Keywords

Comments

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
A finite sequence is periodic if its cyclic rotations are not all different.

Examples

			The a(n) partitions for n = 2, 5, 8, 14, 16, 22:
  11  32     53        95              5533              7744
      11111  3221      5432            7441              9652
             11111111  32222111        533311            554332
                       11111111111111  33222211          54333211
                                       1111111111111111  332222221111
                                                         1111111111111111111111

Crossrefs

The Heinz numbers of these partitions are given by A329132.
The aperiodic version is A329136.
The non-augmented version is A329144.
Periodic binary words are A152061.
Periodic compositions are A178472.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is periodic are A329140.
Cf. A000740, A027375, A328594, A329133, A329135, A325351, A325356, A329134.

Programs

  • Mathematica
    aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
aug[y_]:=Table[If[i

Formula

a(n) + A329136(n) = A000041(n).

Extensions

More terms from Jinyuan Wang, Jun 27 2020

A329144 Number of integer partitions of n whose differences are a periodic word.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 1, 2, 5, 3, 2, 8, 2, 5, 9, 7, 5, 12, 7, 7, 19, 9, 9, 21, 12, 15, 23, 18, 17, 29, 21, 19, 42, 23, 31, 42, 38, 29, 53, 43, 44, 62, 49, 52, 79, 55, 72, 75, 87, 63, 117, 79, 104, 107, 120, 99, 156, 117, 143, 147
Offset: 1

Views

Author

Gus Wiseman, Nov 10 2019

Keywords

Comments

A finite sequence is periodic if its cyclic rotations are not all different.

Examples

			The a(n) partitions for n = 3, 6, 8, 9, 12, 15, 16:
  111  222     2222      333        444           555              4444
       321     11111111  432        543           654              7531
       111111            531        642           753              44332
                         32211      741           852              3332221
                         111111111  3333          951              4332211
                                    222222        33333            22222222
                                    3222111       54321            1111111111111111
                                    111111111111  322221111
                                                  111111111111111

Crossrefs

The Heinz numbers of these partitions are given by A329134.
The augmented version is A329143.
Periodic binary words are A152061.
Periodic compositions are A178472.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is periodic are A329140.
Cf. A000740, A027375, A059966, A328594, A328596, A329132, A329135, A329136.

Programs

  • Mathematica
    aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Table[Length[Select[IntegerPartitions[n],!aperQ[Differences[#]]&]],{n,30}]

A329137 Number of integer partitions of n whose differences are an aperiodic word.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 14, 20, 25, 39, 54, 69, 99, 130, 167, 224, 292, 373, 483, 620, 773, 993, 1246, 1554, 1946, 2421, 2987, 3700, 4548, 5575, 6821, 8330, 10101, 12287, 14852, 17935, 21599, 25986, 31132, 37295, 44539, 53112, 63212, 75123, 89055, 105503, 124682
Offset: 0

Views

Author

Gus Wiseman, Nov 09 2019

Keywords

Comments

A sequence is aperiodic if its cyclic rotations are all different.

Examples

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)    (3)    (4)      (5)        (6)          (7)
       (1,1)  (2,1)  (2,2)    (3,2)      (3,3)        (4,3)
                     (3,1)    (4,1)      (4,2)        (5,2)
                     (2,1,1)  (2,2,1)    (5,1)        (6,1)
                              (3,1,1)    (4,1,1)      (3,2,2)
                              (2,1,1,1)  (2,2,1,1)    (3,3,1)
                                         (3,1,1,1)    (4,2,1)
                                         (2,1,1,1,1)  (5,1,1)
                                                      (2,2,2,1)
                                                      (3,2,1,1)
                                                      (4,1,1,1)
                                                      (2,2,1,1,1)
                                                      (3,1,1,1,1)
                                                      (2,1,1,1,1,1)
With differences:
  ()  ()   ()   ()     ()       ()         ()
      (0)  (1)  (0)    (1)      (0)        (1)
                (2)    (3)      (2)        (3)
                (1,0)  (0,1)    (4)        (5)
                       (2,0)    (3,0)      (0,2)
                       (1,0,0)  (0,1,0)    (1,0)
                                (2,0,0)    (2,1)
                                (1,0,0,0)  (4,0)
                                           (0,0,1)
                                           (1,1,0)
                                           (3,0,0)
                                           (0,1,0,0)
                                           (2,0,0,0)
                                           (1,0,0,0,0)

Crossrefs

The Heinz numbers of these partitions are given by A329135.
The periodic version is A329144.
The augmented version is A329136.
Aperiodic binary words are A027375.
Aperiodic compositions are A000740.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is aperiodic are A329139.
Cf. A152061, A325356, A329132, A329133, A329134, A329140.

Programs

  • Mathematica
    aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Table[Length[Select[IntegerPartitions[n],aperQ[Differences[#]]&]],{n,0,30}]

Formula

a(n) + A329144(n) = A000041(n).

A334298 Numbers whose prime signature is a reversed Lyndon word.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 52, 53, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 84, 88, 89, 92, 96, 97, 99, 101, 103, 104, 107, 109, 112, 113, 116
Offset: 1

Views

Author

Gus Wiseman, Jun 10 2020

Keywords

Comments

A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations.
A number's prime signature is the sequence of positive exponents in its prime factorization.

Examples

			The prime signature of 4200 is (3,1,2,1), which is a reversed Lyndon word, so 4200 is in the sequence.
The sequence of terms together with their prime indices begins:
   1: {}           23: {9}            48: {1,1,1,1,2}
   2: {1}          24: {1,1,1,2}      49: {4,4}
   3: {2}          25: {3,3}          52: {1,1,6}
   4: {1,1}        27: {2,2,2}        53: {16}
   5: {3}          28: {1,1,4}        56: {1,1,1,4}
   7: {4}          29: {10}           59: {17}
   8: {1,1,1}      31: {11}           60: {1,1,2,3}
   9: {2,2}        32: {1,1,1,1,1}    61: {18}
  11: {5}          37: {12}           63: {2,2,4}
  12: {1,1,2}      40: {1,1,1,3}      64: {1,1,1,1,1,1}
  13: {6}          41: {13}           67: {19}
  16: {1,1,1,1}    43: {14}           68: {1,1,7}
  17: {7}          44: {1,1,5}        71: {20}
  19: {8}          45: {2,2,3}        72: {1,1,1,2,2}
  20: {1,1,3}      47: {15}           73: {21}

Crossrefs

The non-reversed version is A329131.
Lyndon compositions are A059966.
Prime signature is A124010.
Numbers with strictly decreasing prime multiplicities are A304686.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose prime signature is a necklace are A329138.
Numbers whose prime signature is aperiodic are A329139.
Cf. A000031, A000740, A000961, A001037, A025487, A027375, A097318, A112798, A118914, A304678, A318731, A329140, A329142.

Programs

  • Mathematica
    lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
Select[Range[100],lynQ[Reverse[Last/@If[#==1,{},FactorInteger[#]]]]&]
Previous Showing 11-14 of 14 results.

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