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%I #10 Jan 09 2013 15:58:56
%S 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 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 93,2,74,7,113,0,100,8,131,0,128
%N 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePartition.html">Prime Partition.</a>
%e 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 Thus a(17)=3.
%t ConjugatePartition[l_List] :=
%t Module[{i, r = Reverse[l], n = Length[l]},
%t Table[n + 1 - Position[r, _?(# >= i &), Infinity, 1][[1, 1]], {i,
%t l[[1]]}]];f[n_] := Select[Select[IntegerPartitions[n], And @@ (PrimeQ[#]) &],
%t And @@ (PrimeQ[ConjugatePartition[#]]) &];a[n_] := Length[f[n]];Table[a[n],{n,1,40}]
%Y Cf. A000040, A000041, A000607
%K nonn
%O 1,6
%A _Ben Branman_, Jan 09 2013