A353401 Number of integer compositions of n with all prime run-lengths.

1, 0, 1, 1, 1, 1, 4, 3, 6, 9, 10, 18, 27, 35, 54, 83, 107, 176, 242, 354, 515, 774, 1070, 1648, 2332, 3429, 4984, 7326, 10521, 15591, 22517, 32908, 48048, 70044, 101903, 149081, 216973, 316289, 461959, 672664, 981356, 1431256, 2086901, 3041577, 4439226, 6467735

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Example

The a(0) = 1 through a(9) = 9 compositions (empty column indicated by dot, 0 is the empty composition):
  0   .  11   111   22   11111   33     11122     44       333
                                 222    22111     1133     11133
                                 1122   1111111   3311     33111
                                 2211             11222    111222
                                                  22211    222111
                                                  112211   1111122
                                                           1112211
                                                           1122111
                                                           2211111

Maple

b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h, add(
     `if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=1..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, May 18 2022

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MemberQ[Length/@Split[#], _?(!PrimeQ[#]&)]&]], {n, 0, 15}]

Crossrefs

The case of runs equal to 2 is A003242 aerated.

The <= 1 version is A003242 ranked by A333489.

The version for parts instead of run-lengths is A023360, both A353429.

The version for partitions is A055923.

The > 1 version is A114901, ranked by A353427.

The <= 2 version is A128695, matching A335464.

The > 2 version is A353400, partitions A100405.

Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351013, A351017.

A005811 counts runs in binary expansion.

A008466 counts compositions with some part > 2.

A011782 counts compositions.

A167606 counts compositions with adjacent parts coprime.

A329738 counts uniform compositions, partitions A047966.

Cf. A078012, A165413, A175413, A274174, A333381, A333755, A353390, A353391, A353392, A353402, A353403.

Sequence in context: A016702 A198610 A128754 * A196889 A005522 A276202

Adjacent sequences: A353398 A353399 A353400 * A353402 A353403 A353404

Keyword

nonn

Author

Gus Wiseman, May 15 2022

Extensions

a(21)-a(45) from Alois P. Heinz, May 18 2022

Status

approved