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A353401
Number of integer compositions of n with all prime run-lengths.
9
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
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.
Sequence in context: A016702 A198610 A128754 * A196889 A005522 A276202
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 15 2022
EXTENSIONS
a(21)-a(45) from Alois P. Heinz, May 18 2022
STATUS
approved