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 A308809 #10 Sep 28 2024 15:59:28
%S A308809 0,0,0,0,0,0,0,0,8,9,10,22,24,26,42,30,48,51,72,76,120,63,132,115,168,
%T A308809 125,234,135,308,203,330,217,416,198,476,315,540,296,684,351,840,410,
%U A308809 798,473,1056,450,1196,564,1248,637,1500,612,1768,795,1782,880
%N A308809 Sum of all the parts in the partitions of n into 4 primes.
%H A308809 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A308809 a(n) = n * Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} c(i) * c(j) * c(k) * c(n-i-j-k), where c = A010051.
%F A308809 a(n) = n * A259194(n).
%F A308809 a(n) = A308771(n) + A308772(n) + A308773(n) + A308774(n).
%t A308809 Table[n*Sum[Sum[Sum[(PrimePi[k] - PrimePi[k - 1])*(PrimePi[j] - PrimePi[j - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
%t A308809 Table[Total[Flatten[Select[IntegerPartitions[n,{4}],AllTrue[#,PrimeQ]&]]],{n,0,60}] (* _Harvey P. Dale_, Sep 28 2024 *)
%Y A308809 Cf. A010051, A259194, A308771, A308772, A308773, A308774.
%K A308809 nonn
%O A308809 0,9
%A A308809 _Wesley Ivan Hurt_, Jun 25 2019