A331901 Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.

1, 1, 3, 3, 1, 3, 25, 9, 61, 91, 99, 151, 901, 303, 1759, 3379, 5239, 4713, 8227, 12901, 12537, 23059, 65239, 159421, 232369, 489817, 351237, 726295, 564363, 1101883, 2517865, 6916027, 11825821, 4942227, 27166753, 21280053, 39547957, 52630273, 113638975

Offset: 1

Author

Ilya Gutkovskiy, Jan 31 2020

Keywords

• nonn

Examples

			a(4) = 3 because we have [7], [5, 2] and [2, 5].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to compositions

Crossrefs

Cf. A000040, A023360, A056768, A070215, A219107, A265112, A299168.

Programs

Maple

s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
    end:
a:= n-> b(ithprime(n), n, 0):
seq(a(n), n=1..42);  # Alois P. Heinz, Jan 31 2020

Mathematica

s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n - 1]];
b[n_, i_, t_] := b[n, i, t] = If[s[i] < n, 0, If[n == 0, t!, Function[p, If[p > n, 0, b[n - p, i - 1, t + 1]]][Prime[i]] + b[n, i - 1, t]]];
a[n_] := b[Prime[n], n, 0];
Array[a, 42] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

Formula

a(n) = A219107(A000040(n)).