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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
OFFSET
1,4
LINKS
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
Sequence in context: A361848 A008768 A067593 * A046966 A225973 A329165
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Jul 17 2003
STATUS
approved