OFFSET
1,5
COMMENTS
Conjecture: a(n)>0 for all n>7.
This has been verified for n up to 3*10^8.
Zhi-Wei Sun also made the following general conjecture: For each odd integer m not congruent to 5 modulo 6, any sufficiently large integer n can be written as p+q with p and 2*p*q+m both prime.
For example, when m = 3, -3, 7, 9, -9, -11, 13, 15, it suffices to require that n is greater than 1, 29, 16, 224, 29, 5, 10, 52 respectively.
Sun also guessed that any integer n>4190 can be written as p+q with p, 2*p*q+1, 2*p*q+7 all prime, and any even number n>1558 can be written as p+q with p, q, 2*p*q+3 all prime. He has some other similar observations.
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=3+7=7+3 with 2*3*7+1=43 prime.
a(263)=1 since 83 is the only prime p with 2p(263-p)+1 prime.
MATHEMATICA
a[n_]:=a[n]=Sum[If[PrimeQ[2Prime[k](n-Prime[k])+1]==True, 1, 0], {k, 1, PrimePi[n]}]
Do[Print[n, " ", a[n]], {n, 1, 1000}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 30 2012
STATUS
approved