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

2,3

Comments

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

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A023360, A121303.

Sequence in context: A154696 A154698 A063786 * A002106 A232316 A184604

Adjacent sequences: A121301 A121302 A121303 * A121305 A121306 A121307

Keyword

nonn

Author

Emeric Deutsch, Aug 06 2006

Status

approved