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

1,2

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.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A190579 A271527 A214543 * A370557 A024033 A255569

Adjacent sequences: A258658 A258659 A258660 * A258662 A258663 A258664

Keyword

nonn

Author

Zhi-Wei Sun, Jun 06 2015

Status

approved