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 A202177 #11 Feb 16 2025 08:33:16
%S A202177 0,0,0,1,0,2,0,1,1,2,0,2,2,2,2,3,3,0,4,2,5,2,4,3,8,2,6,4,11,0,10,4,14,
%T A202177 2,14,4,21,2,20,5,25,0,28,6,30,2,38,5,46,0,44,4,54,0,56,6,67,2,72,4,
%U A202177 93,2,74,7,113,0,100,8,131,0,128
%N A202177 Number of partitions p of n such that each part of p is prime and each part of the conjugate partition of p is also prime.
%H A202177 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePartition.html">Prime Partition.</a>
%e A202177 For n=17, there are three valid partitions: (7,7,3), its conjugate partition (3,3,3,2,2,2,2), and the self-conjugate partition (5,5,3,2,2).
%e A202177 Thus a(17)=3.
%t A202177 ConjugatePartition[l_List] :=
%t A202177 Module[{i, r = Reverse[l], n = Length[l]},
%t A202177 Table[n + 1 - Position[r, _?(# >= i &), Infinity, 1][[1, 1]], {i,
%t A202177 l[[1]]}]];f[n_] := Select[Select[IntegerPartitions[n], And @@ (PrimeQ[#]) &],
%t A202177 And @@ (PrimeQ[ConjugatePartition[#]]) &];a[n_] := Length[f[n]];Table[a[n],{n,1,40}]
%Y A202177 Cf. A000040, A000041, A000607
%K A202177 nonn
%O A202177 1,6
%A A202177 _Ben Branman_, Jan 09 2013