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

0,8

Links

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

Index entries for sequences related to compositions

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... .

Example

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).

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]

Crossrefs

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

Sequence in context: A355638 A027069 A238494 * A325550 A362608 A374688

Adjacent sequences: A359385 A359386 A359387 * A359389 A359390 A359391

Keyword

nonn

Author

Stefano Spezia, Dec 29 2022

Status

approved