A121304 Number of parts in all the compositions of n into primes (i.e., in all ordered sequences of primes having sum n).

1, 1, 2, 5, 5, 14, 17, 32, 53, 76, 139, 198, 334, 515, 798, 1280, 1938, 3075, 4710, 7299, 11298, 17296, 26738, 40874, 62763, 96036, 146674, 224210, 341562, 520767, 792375, 1204951, 1831124, 2779234, 4217008, 6391663, 9683056, 14659038, 22177341

Offset: 2

Author

Emeric Deutsch, Aug 06 2006

Keywords

• nonn

Comments

a(n) = Sum_{k=1..floor(n/2)} k*A121303(n,k).

Examples

			a(8) = 17 because the compositions of 8 into primes are [3,5], [5,3], [2,3,3], [3,2,3], [3,3,2] and [2,2,2,2], having a total of 2+2+3+3+3+4 = 17 parts.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..2000

Crossrefs

Cf. A023360, A121303.

Programs

Maple

g:=sum(z^ithprime(i),i=1..53)/(1-sum(z^ithprime(i),i=1..53))^2: gser:=series(g,z=0,48): seq(coeff(gser,z,n),n=2..45);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      `if`(isprime(j), (p->p+[0, p[1]])(b(n-j)), 0), j=1..n))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=2..50);  # Alois P. Heinz, Nov 08 2013, revised Feb 12 2021

Mathematica

nn=40;a[x_]:=Sum[x^Prime[n],{n,1,nn}];Drop[CoefficientList[Series[D[1/(1-y a[x]),y]/.y ->1,{x,0,nn}],x],2] (* Geoffrey Critzer, Nov 08 2013 *)
Table[Length[Flatten[Union[Flatten[Permutations/@Select[ IntegerPartitions[ n], AllTrue[ #,PrimeQ]&],1]]]],{n,2,40}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 24 2016 *)
b[n_] := b[n] = If[n == 0, {1, 0}, Sum[If[PrimeQ[j],
     Function[p, p+{0, p[[1]]}][b[n-j]], {0, 0}], {j, 1, n}]];
a[n_] := b[n][[2]];
a /@ Range[2, 50] (* Jean-François Alcover, Jun 01 2021, after Alois P. Heinz *)

Formula

G.f.: (Sum_{i>=1} z^prime(i))/(1 - Sum_{i>=1} z^prime(i))^2.