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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 06 2006
STATUS
approved