A359388 a(n) is the number of compositions of n into prime parts, with the 1st part equal to 2, the 2nd part less than or equal to 3, ..., and the k-th part less than or equal to prime(k), and so on.

1, 0, 1, 0, 1, 1, 1, 2, 2, 4, 5, 7, 11, 15, 24, 33, 50, 73, 105, 159, 229, 342, 501, 738, 1094, 1604, 2378, 3499, 5166, 7627, 11243, 16610, 24494, 36165, 53376, 78775, 116301, 171642, 253398, 374034, 552139, 815079, 1203166, 1776174, 2621938, 3870572, 5713798, 8434744

Offset: 0

Author

Stefano Spezia, Dec 29 2022

Keywords

• nonn

Examples

			The 7 such compositions of n = 11 are:
[ 1]  (2, 2, 2, 2, 3);
[ 2]  (2, 2, 2, 3, 2);
[ 3]  (2, 2, 3, 2, 2);
[ 4]  (2, 3, 2, 2, 2);
[ 5]  (2, 2, 2, 5);
[ 6]  (2, 2, 5, 2);
[ 7]  (2, 3, 3, 3).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4000
• Index entries for sequences related to compositions

Crossrefs

Cf. A000040, A004526, A023360, A078974, A326178, A359328.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      b(n-ithprime(j), i+1), j=1..min(i, numtheory[pi](n))))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 29 2022

Mathematica

a[n_]:=Coefficient[Expand[Sum[Product[Sum[x^Prime[i], {i, k}], {k,m}], {m, 0,Floor[n/2]}]],x,n]; Array[a,48,0]

Formula

G.f.: Sum_{m>=0} Product_{k=1..m} Sum_{i=1..k} x^prime(i).

a(n) ~ c*A078974^n, where c = 0.094587447... .