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%I #12 Oct 01 2021 15:37:36
%S 0,0,0,0,0,0,0,0,0,0,2,2,3,5,6,8,11,11,13,20,20,27,28,35,32,51,41,60,
%T 53,74,75,112,83,136,103,162,126,205,143,246,170,283,219,365,217,415,
%U 276,475,310,554,320,642,376,690,446,835,443,944,532,1019,587
%N Sum of the second largest parts in the partitions of n into 5 primes.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l) * i, where c = A010051.
%F a(n) = A308854(n) - A308855(n) - A308856(n) - A308857(n) - A308859(n).
%t Table[Total[Select[IntegerPartitions[n,{5}],AllTrue[#,PrimeQ]&][[All,2]]],{n,0,60}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Mar 15 2020 *)
%Y Cf. A010051, A259195, A308854, A308855, A308856, A308857, A308859.
%K nonn
%O 0,11
%A _Wesley Ivan Hurt_, Jun 28 2019