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

Author

Gus Wiseman, May 16 2022

Keywords

• nonn

Examples

			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)
		

Links

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

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.

Programs

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}]

Extensions

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