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 A308923 #11 Sep 24 2024 19:07:22
%S A308923 0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,5,5,8,10,9,11,16,18,21,28,27,36,41,48,
%T A308923 46,67,54,82,78,99,86,126,104,156,129,181,152,238,175,277,221,325,249,
%U A308923 405,295,480,342,542,394,660,430,752,517,851,584,1005,643
%N A308923 Sum of the third largest parts in the partitions of n into 6 primes.
%H A308923 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A308923 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)} c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m) * j, where c = A010051.
%F A308923 a(n) = A308919(n) - A308920(n) - A308921(n) - A308922(n) - A308924(n) - A308925(n).
%t A308923 Table[Sum[Sum[Sum[Sum[Sum[j*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[n - i - j - k - l - m] - PrimePi[n - i - j - k - l - m - 1]), {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]
%t A308923 Table[Total[Select[IntegerPartitions[n,{6}],AllTrue[#,PrimeQ]&][[;;,3]]],{n,-0,70}] (* _Harvey P. Dale_, Sep 24 2024 *)
%Y A308923 Cf. A010051, A259196, A308919, A308920, A308921, A308922, A308924, A308925.
%K A308923 nonn
%O A308923 0,13
%A A308923 _Wesley Ivan Hurt_, Jun 30 2019