This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A308857 #12 Oct 01 2021 15:38:48
%S A308857 0,0,0,0,0,0,0,0,0,0,2,2,2,5,5,8,7,9,8,16,12,21,20,27,20,41,27,46,34,
%T A308857 54,44,78,44,86,59,104,65,129,79,152,96,175,115,221,117,249,157,295,
%U A308857 170,342,179,394,214,430,243,517,245,584,307,643,332,730,352
%N A308857 Sum of the third largest parts in the partitions of n into 5 primes.
%H A308857 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A308857 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) * j, where c = A010051.
%F A308857 a(n) = A308854(n) - A308855(n) - A308856(n) - A308858(n) - A308859(n).
%t A308857 Table[Total[Select[IntegerPartitions[n,{5}],AllTrue[#,PrimeQ]&][[All,3]]],{n,0,70}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 06 2020 *)
%Y A308857 Cf. A010051, A259195, A308854, A308855, A308856, A308858, A308859.
%K A308857 nonn
%O A308857 0,11
%A A308857 _Wesley Ivan Hurt_, Jun 28 2019