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 A084993 Total number of parts in all partitions of n into prime parts. 13
 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 (list; graph; refs; listen; history; text; internal format)
 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 *) 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: A026317 A008768 A067593 * A046966 A225973 A329165 Adjacent sequences:  A084990 A084991 A084992 * A084994 A084995 A084996 KEYWORD nonn AUTHOR Vladeta Jovovic, Jul 17 2003 STATUS approved

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Last modified August 4 00:42 EDT 2020. Contains 336201 sequences. (Running on oeis4.)