A258141 Number of ways to write n as p^2 + q with p, q and 2*p + 3 all prime.

0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0

Offset: 1

Author

Zhi-Wei Sun, May 22 2015

Keywords

• nonn

Comments

Conjecture: For any integer n > 0, we have a(n+r) > 0 for some r = 0,1,2,3,4,5.

We have verified this for n up to 10^8. See also A258139 for a weaker version of this conjecture.

The conjecture is somewhat similar to Goldbach's Conjecture. It implies that there are infinitely many primes p with 2*p + 3 prime.

Examples

			a(11) = 1 since 11 = 2^2 + 7 with 2, 7 and 2*2 + 3 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A002375, A023204, A258139, A258140.

Programs

Mathematica

Do[r=0;Do[If[PrimeQ[2Prime[k]+3]&&PrimeQ[n-Prime[k]^2],r=r+1],{k,1,PrimePi[Sqrt[n]]}];Print[n," ",r];Continue,{n,1,100}]