A084993 Total number of parts in all partitions of n into prime parts.

0, 1, 1, 2, 3, 5, 6, 9, 12, 16, 20, 27, 33, 42, 53, 64, 80, 96, 117, 141, 169, 201, 239, 282, 333, 390, 456, 532, 617, 715, 826, 951, 1094, 1253, 1435, 1636, 1864, 2119, 2404, 2723, 3078, 3473, 3915, 4403, 4947, 5549, 6215, 6952, 7767, 8665, 9656, 10748

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

G.f.: sum(x^p(j)/(1-x^p(j)),j=1..infinity)/product(1-x^p(j), j=1..infinity), where p(j) is the j-th prime. - Emeric Deutsch, Mar 07 2006

Example

Partitions of 9 into primes are 2+2+2+3=3+3+3=2+2+5=2+7; thus a(9)=4+3+3+2=12.

Maple

g:=sum(x^ithprime(j)/(1-x^ithprime(j)), j=1..20)/product(1-x^ithprime(j), j=1..20): gser:=series(g, x=0, 60): seq(coeff(gser, x^n), n=1..57); # Emeric Deutsch, Mar 07 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]
    elif i=1 then `if`(irem(n, 2)=0, [1, n/2], [0, 0])
    else g:= `if`(ithprime(i)>n, [0$2], b(n-ithprime(i), i));
         b(n, i-1) +g +[0, g[1]]
      fi
    end:
a:= n-> b(n, pi(n))[2]:
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 30 2012

Mathematica

nn=40; a=Product[1/(1-y x^i), {i, Table[Prime[n], {n, 1, nn}]}]; Drop[CoefficientList[Series[D[a, y]/.y->1, {x, 0, nn}], x], 1]  (* Geoffrey Critzer, Oct 30 2012 *)
b[n_, i_] := b[n, i] = Module[{g}, Which[n == 0, {1, 0}, i < 1, {0, 0}, i == 1, If[EvenQ[n], {1, n/2}, {0, 0}], True, g = If[Prime[i] > n, {0, 0}, b[n - Prime[i], i]]; b[n, i - 1] + g + {0, g[[1]]}]];
a[n_] := b[n, PrimePi[n]][[2]];
Array[a, 52] (* Jean-François Alcover, Dec 30 2017, after Alois P. Heinz *)
Table[Length[Flatten[Select[IntegerPartitions[n], AllTrue[#, PrimeQ]&]]], {n, 60}] (* Harvey P. Dale, Jul 11 2023 *)

Prog

(PARI)
sumparts(n, pred)={sum(k=1, n, 1/(1-pred(k)*x^k) - 1 + 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. A000607, A024938.

Sequence in context: A361848 A008768 A067593 * A046966 A225973 A329165

Adjacent sequences: A084990 A084991 A084992 * A084994 A084995 A084996

Keyword

nonn

Author

Vladeta Jovovic, Jul 17 2003

Status

approved