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

1,5

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A084993.

Sequence in context: A229964 A309722 A070298 * A332715 A219107 A338498

Adjacent sequences: A024935 A024936 A024937 * A024939 A024940 A024941

Keyword

easy,nonn

Author

Clark Kimberling

Extensions

More terms from Vladeta Jovovic, Jul 17 2003

Status

approved