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A180041 Number of Goldbach partitions of (2n)^n.

%I A180041 #14 Apr 15 2019 05:23:45
%S A180041 0,2,13,53,810,20564,274904,6341424,419586990
%N A180041 Number of Goldbach partitions of (2n)^n.
%C A180041 This is the main diagonal of the array mentioned in A180007, only considering even rows (as odd numbers cannot be the sums of two odd primes), namely A(2n, n) = number of ways of writing (2n)^n as the sum of two odd primes, when the order does not matter.
%F A180041 a(n) = A061358((2*n)^n) = A061358(A062971(n)).
%e A180041 a(1) = 0 because 2*1 = 2 is too small to be the sum of two primes.
%e A180041 a(2) = 2 because 4^2 = 16 = 3+13 = 5+11.
%e A180041 a(3) = 13 because 6^3 = 216 and A180007(3) = Number of Goldbach partitions of 6^3 = 13.
%e A180041 a(4) = 53 because 8^4 = 2^12 and A006307(12) = Number of ways writing 2^12 as unordered sums of 2 primes.
%p A180041 A180041 := proc(n) local a,m,p: if(n=1)then return 0:fi: a:=0: m:=(2*n)^n: p:=prevprime(ceil((m-1)/2)): while p > 2 do if isprime(m-p) then a:=a+1: fi: p := prevprime(p): od: return a: end: seq(A180041(n),n=1..5); # _Nathaniel Johnston_, May 08 2011
%t A180041 f[n_] := Block[{c = 0, p = 3, m = (2 n)^n}, lmt = Floor[m/2] + 1; While[p < lmt, If[ PrimeQ[m - p], c++ ]; p = NextPrime@p]; c]; Do[ Print[{n, f@n // Timing}], {n, 8}] (* _Robert G. Wilson v_, Aug 10 2010 *)
%Y A180041 Cf. A001031, A061358, A065577, A180007.
%K A180041 more,nonn
%O A180041 1,2
%A A180041 _Jonathan Vos Post_, Aug 07 2010
%E A180041 a(6)-a(8) from _Robert G. Wilson v_, Aug 10 2010
%E A180041 a(9) from _Giovanni Resta_, Apr 15 2019