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

0, 2, 13, 53, 810, 20564, 274904, 6341424, 419586990

Offset: 1

Author

Jonathan Vos Post, Aug 07 2010

Keywords

• more
• nonn

Comments

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.

Examples

			a(1) = 0 because 2*1 = 2 is too small to be the sum of two primes.
a(2) = 2 because 4^2 = 16 = 3+13 = 5+11.
a(3) = 13 because 6^3 = 216 and A180007(3) = Number of Goldbach partitions of 6^3 = 13.
a(4) = 53 because 8^4 = 2^12 and A006307(12) = Number of ways writing 2^12 as unordered sums of 2 primes.
		

Crossrefs

Cf. A001031, A061358, A065577, A180007.

Programs

Maple

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

Mathematica

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 *)

Formula

a(n) = A061358((2*n)^n) = A061358(A062971(n)).

Extensions

a(6)-a(8) from Robert G. Wilson v, Aug 10 2010

a(9) from Giovanni Resta, Apr 15 2019