A258661 Positive integers m such that none of the four consecutive numbers m, m+1, m+2, m+3 can be written as p^2 + q with p and q both prime.

1, 2, 1009, 3598, 4354, 9214, 11662, 15051, 15052, 15873, 15874, 19042, 21772, 22497, 22498, 24334, 26242, 46654, 60514, 76173, 76174, 93634, 97341, 97342, 108898, 112893, 112894, 121101, 121102, 133954, 152098, 156813, 156814, 166462, 171393, 171394, 181473, 181474, 202498, 213441, 213442, 224674, 236193, 236194, 254013, 254014, 266253, 266254, 272482, 278781

Offset: 1

Author

Zhi-Wei Sun, Jun 06 2015

Keywords

• nonn

Comments

Conjecture: Any term not among 1, 2, 1009, 15051, 15052, 21772 has the form 36*k^2-2 or the form 36*k^2-3, where k is a positive integer.

Note that this conjecture implies the conjecture in A258168 since neither 36*k^2-2 nor 36*k^2-3 can be a multiple of 5.

For more comments, see the linked message to Number Theory Mailing List.

Examples

			a(1) = 1 since none of 1,2,3,4 has the form p^2+q with p and q both prime.
a(2) = 2 since none of 2,3,4,5 has the form p^2+q with p and q both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..500
• Zhi-Wei Sun, Re: A new conjecture involving p^2+q, a message to Number Theory Mailing List, May 30, 2015.

Crossrefs

Cf. A000040, A258139, A258140, A258141, A258168.

Programs

Mathematica

n=0;Do[Do[If[PrimeQ[m+r-Prime[k]^2],Goto[aa]],{r,0,3},{k,1,PrimePi[Sqrt[m+r]]}];n=n+1;Print[n, " ", m];Label[aa];Continue,{m,1,278781}]