A359388 - OEIS

%I #33 Jan 01 2023 14:49:39

%S 1,0,1,0,1,1,1,2,2,4,5,7,11,15,24,33,50,73,105,159,229,342,501,738,

%T 1094,1604,2378,3499,5166,7627,11243,16610,24494,36165,53376,78775,

%U 116301,171642,253398,374034,552139,815079,1203166,1776174,2621938,3870572,5713798,8434744

%N 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.

%H Alois P. Heinz, <a href="/A359388/b359388.txt">Table of n, a(n) for n = 0..4000</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

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

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

%e The 7 such compositions of n = 11 are:

%e [ 1]  (2, 2, 2, 2, 3);

%e [ 2]  (2, 2, 2, 3, 2);

%e [ 3]  (2, 2, 3, 2, 2);

%e [ 4]  (2, 3, 2, 2, 2);

%e [ 5]  (2, 2, 2, 5);

%e [ 6]  (2, 2, 5, 2);

%e [ 7]  (2, 3, 3, 3).

%p b:= proc(n, i) option remember; `if`(n=0, 1, add(

%p       b(n-ithprime(j), i+1), j=1..min(i, numtheory[pi](n))))

%p     end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Dec 29 2022

%t 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]

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

%K nonn

%O 0,8

%A _Stefano Spezia_, Dec 29 2022