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%I #46 Sep 08 2022 08:46:13
%S 1,1,1,2,2,3,4,4,5,7,7,9,10,11,12,16,16,20,21,24,26,33,31,39,39,47,46,
%T 59,53,69,65,80,77,98,85,114,104,131,118,154,133,179,155,200,177,236,
%U 196,268,227,300,256
%N Number of partitions of n into nine primes.
%H Robert Israel, <a href="/A259200/b259200.txt">Table of n, a(n) for n = 18..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = [x^n y^9] Product_{k>=1} 1/(1 - y*x^prime(k)). - _Ilya Gutkovskiy_, Apr 18 2019
%F a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} A010051(q) * A010051(p) * A010051(o) * A010051(m) * A010051(l) * A010051(k) * A010051(j) * A010051(i) * A010051(n-i-j-k-l-m-o-p-q). - _Wesley Ivan Hurt_, Jul 13 2019
%e a(23) = 3 because there are 3 partitions of 23 into nine primes: [2,2,2,2,2,2,2,2,7], [2,2,2,2,2,2,3,3,5] and [2,2,2,2,3,3,3,3,3].
%p N:= 100: # to get a(0) to a(N)
%p Primes:= select(isprime,[$1..N]):
%p np:= nops(Primes):
%p for j from 0 to np do g[0,j]:= 1 od:
%p for n from 1 to 9 do
%p g[n,0]:= 0:
%p for j from 1 to np do
%p g[n,j]:= convert(series(add(g[k,j-1]
%p *x^((n-k)*Primes[j]),k=0..n),x,N+1),polynom)
%p od
%p od:
%p seq(coeff(g[9,np],x,i),i=18..N) # _Robert Israel_, Jun 21 2015
%t Table[Length[Select[IntegerPartitions[n],Length[#]==9&&AllTrue[ #, PrimeQ]&]], {n,18,70}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Jul 31 2016 *)
%o (PARI) a(n) = {nb = 0; forpart(p=n, if (#p && (#select(x->isprime(x), Vec(p)) == #p), nb+=1), , [9,9]); nb;} \\ _Michel Marcus_, Jun 21 2015
%o (Magma) [#RestrictedPartitions(k,9,Set(PrimesUpTo(1000))):k in [18..70]] ; // _Marius A. Burtea_, Jul 13 2019
%Y Column k=9 of A117278.
%Y Number of partitions of n into r primes for r = 1..10: A010051, A061358, A068307, A259194, A259195, A259196, A259197, A259198, this sequence, A259201.
%Y Cf. A000040.
%K nonn,easy
%O 18,4
%A _Doug Bell_, Jun 20 2015
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