A024938 Total number of parts in all partitions of n into distinct prime parts.

0, 1, 1, 0, 3, 0, 3, 2, 2, 5, 1, 5, 3, 5, 5, 7, 5, 10, 6, 10, 12, 10, 15, 12, 16, 17, 17, 19, 22, 17, 27, 21, 30, 30, 31, 35, 36, 40, 45, 45, 49, 53, 50, 62, 60, 69, 69, 73, 78, 85, 88, 98, 100, 105, 116, 116, 134, 135, 141, 149, 154, 168, 176, 188, 195, 206, 211, 232, 242, 255, 267, 276

Offset: 1

Author

Clark Kimberling

Keywords

• easy
• nonn

Examples

			a(16) = 7 because the partitions of 16 into distinct prime parts are [13,3], [11,5] and [11,3,2].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A084993.

Programs

Maple

g:=sum(x^ithprime(j)/(1+x^ithprime(j)),j=1..30)*product(1+x^ithprime(j),j=1..30): gser:=series(g,x=0,80): seq(coeff(gser,x,n),n=1..75); # Emeric Deutsch, Apr 01 2006
# second Maple program:
with(numtheory):
b:= proc(n, i) option remember; local g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= `if`(ithprime(i)>n, [0$2], b(n-ithprime(i), i-1));
         b(n, i-1) +g +[0, g[1]]
      fi
    end:
a:= n-> b(n, pi(n))[2]:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 30 2012

Mathematica

Rest@ CoefficientList[ Series[ Sum[x^Prime@j/(1 + x^Prime@j), {j, 20}]* Product[1 + x^Prime@j, {j, 20}], {x, 0, 70}], x] (* Robert G. Wilson v *)
b[n_, i_] := b[n, i] = Module[{g}, If[n==0, {1, 0}, If[i < 1, {0, 0}, g = If[ Prime[i] > n, {0, 0}, b[n - Prime[i], i-1]]; b[n, i-1] + g + {0, g[[1]]}]]]; a[n_] := b[n, PrimePi[n]][[2]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Dec 27 2015, after Alois P. Heinz *)

PARI

sumparts(n, pred)={sum(k=1, n, 1 - 1/(1+pred(k)*x^k) + O(x*x^n))*prod(k=1, n, 1+pred(k)*x^k + O(x*x^n))}
{my(n=60); Vec(sumparts(n, isprime), -n)} \\ Andrew Howroyd, Dec 28 2017

Formula

G.f.: sum(x^p(j)/(1+x^p(j)),j>=1)*product(1+x^p(j), j>=1), where p(j) is the j-th prime. - Vladeta Jovovic, Jul 17 2003

Extensions

More terms from Vladeta Jovovic, Jul 17 2003