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%I #30 Sep 08 2022 08:46:13
%S 0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,2,3,3,4,4,6,3,6,5,7,5,9,5,11,7,11,7,13,
%T 6,14,9,15,8,18,9,21,10,19,11,24,10,26,12,26,13,30,12,34,15,33,16,38,
%U 14,41,17,41,16,45,16,50,19,47,21,56,20,61,20,57
%N Number of partitions of n into four primes.
%H Giovanni Resta, <a href="/A259194/b259194.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} A010051(i) * A010051(j) * A010051(k) * A010051(n-i-j-k). - _Wesley Ivan Hurt_, Apr 17 2019
%F a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^prime(k)). - _Ilya Gutkovskiy_, Apr 18 2019
%e a(17) = 3 because 17 can be written as the sum of four primes in exactly three ways: 2+2+2+11, 2+3+5+7 and 2+5+5+5.
%t a[n_] := Length@ IntegerPartitions[n, {4}, Prime@ Range@ PrimePi@ n]; a /@
%t Range[0, 100] (* _Giovanni Resta_, Jun 21 2015 *)
%t Table[Count[IntegerPartitions[n,{4}],_?(AllTrue[#,PrimeQ]&)],{n,0,80}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Feb 03 2019 *)
%o (PARI) a(n) = {nb = 0; forpart(p=n, if (#p && (#select(x->isprime(x), Vec(p)) == #p), nb+=1), , [4,4]); nb;} \\ _Michel Marcus_, Jun 21 2015
%o (Magma) [0] cat [#RestrictedPartitions(n,4,{d:d in PrimesUpTo(n)}):n in [1..100]]; // _Marius A. Burtea_, May 07 2019
%Y Column k=4 of A117278.
%Y Number of partitions of n into r primes for r = 1..10: A010051, A061358, A068307, this sequence, A259195, A259196, A259197, A259198, A259200, A259201.
%Y Cf. A000040.
%K nonn,easy
%O 0,12
%A _Doug Bell_, Jun 20 2015
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