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Number of partitions of n into six primes.
20

%I #25 Jul 27 2024 15:07:39

%S 1,1,1,2,2,3,4,3,4,5,6,6,8,7,10,10,12,11,16,12,19,17,22,18,26,20,31,

%T 24,33,27,42,29,47,35,51,38,60,41,68,47,73,53,86,54,95,64,103,70,116,

%U 73,131,81,137,89,156,92,171,103,180,112,202,117,223,127,232

%N Number of partitions of n into six primes.

%H Alois P. Heinz, <a href="/A259196/b259196.txt">Table of n, a(n) for n = 12..10000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_(i=j..floor((n-j-k-l-m)/2)} A010051(i) * A010051(j) * A010051(k) * A010051(l) * A010051(m) * A010051(n-i-j-k-l-m). - _Wesley Ivan Hurt_, Apr 17 2019

%F a(n) = [x^n y^6] Product_{k>=1} 1/(1 - y*x^prime(k)). - _Ilya Gutkovskiy_, Apr 18 2019

%e a(17) = 3 because there are 3 partitions of 17 into six primes: [2,2,2,2,2,7], [2,2,2,3,3,5] and [2,3,3,3,3,3].

%t Table[Count[IntegerPartitions[n,{6}],_?(AllTrue[#,PrimeQ]&)],{n,12,80}] (* _Harvey P. Dale_, Jul 27 2024 *)

%Y Column k=6 of A117278.

%Y Number of partitions of n into r primes for r = 1-10: A010051, A061358, A068307, A259194, A259195, this sequence, A259197, A259198, A259200, A259201.

%Y Cf. A000040.

%K nonn,easy

%O 12,4

%A _Doug Bell_, Jun 20 2015