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
Keywords
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..10000
Programs
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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
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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 *) -
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
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