OFFSET
1,6
COMMENTS
Conjecture: a(n)>0 for all n>1. Moreover, any integer n>357 can be written as x+y (x>0, y>0) with 2x*y+1 and 2x*y-1 twin primes.
This conjecture has been verified for n up to 10^8, and it implies the twin prime conjecture.
Zhi-Wei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y, where x and y are positive integers with 2x*y+m and 2x*y-m both prime.
For example, when m=3,5,7,9,11 it suffices to require that n is greater than 5090, 222, 1785, 548, 603 respectively.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
EXAMPLE
a(10)=2 since 10=1+9=3+7 with 2*1*9+1=19 and 2*3*7+1=43 both prime.
MATHEMATICA
a[n_] := a[n] = Sum[If[PrimeQ[2k(n-k)+1] == True, 1, 0], {k, n/2}]; Do[Print[n, " ", a[n]], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 29 2012
STATUS
approved