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 1-3 of 3 results.

A329139 Numbers whose prime signature is an aperiodic word.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104
Offset: 1

Views

Author

Gus Wiseman, Nov 09 2019

Keywords

Comments

First differs from A319161 in having 1260 = 2*2 * 3^2 * 5^1 * 7^1. First differs from A325370 in having 420 = 2^2 * 3^1 * 5^1 * 7^1.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A sequence is aperiodic if its cyclic rotations are all different.

Examples

			The sequence of terms together with their prime signatures begins:
   1: ()
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   7: (1)
   8: (3)
   9: (2)
  11: (1)
  12: (2,1)
  13: (1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  20: (2,1)
  23: (1)
  24: (3,1)
  25: (2)
  27: (3)

Crossrefs

Complement of A329140.
Aperiodic compositions are A000740.
Aperiodic binary words are A027375.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is a necklace are A329138.
Cf. A025487, A097318, A112798, A124010, A178472, A181819, A304678, A329133, A329135, A329136, A329137, A329142.

Programs

  • Mathematica
    aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],aperQ[Last/@FactorInteger[#]]&]

A329135 Numbers whose differences of prime indices are an aperiodic word.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73
Offset: 1

Views

Author

Gus Wiseman, Nov 09 2019

Keywords

Comments

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

Examples

			The sequence of terms together with their differences of prime indices begins:
    1: ()
    2: ()
    3: ()
    4: (0)
    5: ()
    6: (1)
    7: ()
    9: (0)
   10: (2)
   11: ()
   12: (0,1)
   13: ()
   14: (3)
   15: (1)
   17: ()
   18: (1,0)
   19: ()
   20: (0,2)
   21: (2)
   22: (4)

Crossrefs

Complement of A329134.
These are the Heinz numbers of the partitions counted by A329137.
Aperiodic compositions are A000740.
Aperiodic binary words are A027375.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is aperiodic are A329139.
Cf. A056239, A112798, A124010, A152061, A329133, A329136, A329140.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],aperQ[Differences[primeMS[#]]]&]

A329136 Number of integer partitions of n whose augmented differences are an aperiodic word.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 10, 14, 19, 28, 40, 53, 75, 99, 131, 172, 226, 294, 380, 488, 617, 787, 996, 1250, 1565, 1953, 2425, 3003, 3705, 4559, 5589, 6836, 8329, 10132, 12292, 14871, 17950, 21629, 25988, 31169, 37306, 44569, 53139, 63247, 75133, 89111, 105515, 124737
Offset: 0

Views

Author

Gus Wiseman, Nov 09 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 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)
            (2,1)  (2,2)    (4,1)      (3,3)        (4,3)
                   (3,1)    (2,2,1)    (4,2)        (5,2)
                   (2,1,1)  (3,1,1)    (5,1)        (6,1)
                            (2,1,1,1)  (2,2,2)      (3,2,2)
                                       (3,2,1)      (3,3,1)
                                       (4,1,1)      (4,2,1)
                                       (2,2,1,1)    (5,1,1)
                                       (3,1,1,1)    (2,2,2,1)
                                       (2,1,1,1,1)  (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 augmented differences:
  (1)  (2)  (3)    (4)      (5)        (6)          (7)
            (2,1)  (1,2)    (4,1)      (1,3)        (2,3)
                   (3,1)    (1,2,1)    (3,2)        (4,2)
                   (2,1,1)  (3,1,1)    (5,1)        (6,1)
                            (2,1,1,1)  (1,1,2)      (1,3,1)
                                       (2,2,1)      (2,1,2)
                                       (4,1,1)      (3,2,1)
                                       (1,2,1,1)    (5,1,1)
                                       (3,1,1,1)    (1,1,2,1)
                                       (2,1,1,1,1)  (2,2,1,1)
                                                    (4,1,1,1)
                                                    (1,2,1,1,1)
                                                    (3,1,1,1,1)
                                                    (2,1,1,1,1,1)

Crossrefs

The Heinz numbers of these partitions are given by A329133.
The periodic version is A329143.
The non-augmented version is A329137.
Aperiodic binary words are A027375.
Aperiodic compositions are A000740.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose differences of prime indices are aperiodic are A329135.
Numbers whose prime signature is aperiodic are A329139.
Cf. A152061, A325351, A325356, A329132, A329134, A329139, A329140.

Programs

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

Formula

a(n) + A329143(n) = A000041(n).
Showing 1-3 of 3 results.

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