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

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

Offset

1,2

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.

Links

Table of n, a(n) for n=1..9.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A001031, A061358, A065577, A180007.

Sequence in context: A048502 A177077 A144235 * A042061 A229736 A187560

Adjacent sequences: A180038 A180039 A180040 * A180042 A180043 A180044

Keyword

more,nonn

Author

Jonathan Vos Post, Aug 07 2010

Extensions

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

a(9) from Giovanni Resta, Apr 15 2019

Status

approved