A353429 Number of integer compositions of n with all prime parts and all prime run-lengths.

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 4, 0, 2, 2, 5, 4, 9, 1, 5, 12, 20, 11, 19, 18, 31, 43, 54, 37, 63, 95, 121, 124, 154, 178, 261, 353, 393, 417, 565, 770, 952, 1138, 1326, 1647, 2186, 2824, 3261, 3917, 4941, 6423, 7935, 9719, 11554, 14557, 18536, 23380, 27985

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..4000

Example

The a(13) = 2 through a(16) = 9 compositions:
  (22333)  (77)       (555)     (3355)
  (33322)  (2255)     (33333)   (5533)
           (5522)     (222333)  (22255)
           (223322)   (333222)  (55222)
           (2222222)            (332233)
                                (2222233)
                                (2223322)
                                (2233222)
                                (3322222)

Maple

b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h and isprime(i),
      add(`if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, May 18 2022

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]], {n, 0, 15}]

Crossrefs

The first condition only is A023360, partitions A000607.

For partitions we have A351982, only run-lens A100405, only parts A008483.

The second condition only is A353401, partitions A055923.

A003242 counts anti-run compositions, ranked by A333489.

A011782 counts compositions.

A052284 counts compositions into nonprimes, partitions A002095.

A106356 counts compositions by number of adjacent equal parts.

A114901 counts compositions with no runs of length 1, ranked by A353427.

A329738 counts uniform compositions, partitions A047966.

Cf. A005811, A128695, A137200, A167606, A333755, A335464, A353428.

Sequence in context: A109077 A355916 A309023 * A137585 A301344 A301579

Adjacent sequences: A353426 A353427 A353428 * A353430 A353431 A353432

Keyword

nonn

Author

Gus Wiseman, May 16 2022

Extensions

a(26)-a(56) from Alois P. Heinz, May 18 2022

Status

approved