A258153 Numbers of the form p^2 + q with p, q and 2*p + 3 all prime.

6, 7, 9, 11, 15, 17, 21, 23, 27, 28, 30, 32, 33, 35, 36, 38, 41, 42, 44, 45, 47, 48, 51, 52, 54, 56, 57, 60, 62, 63, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 84, 86, 87, 90, 92, 93, 96, 98, 101, 102, 104, 105, 107, 108, 110, 111, 113, 114, 116, 117, 120, 122, 126, 128, 131, 132, 134, 135, 138, 141

Offset: 1

Author

Zhi-Wei Sun, May 22 2015

Keywords

• nonn

Comments

The conjecture in A258141 asserts that any six consecutive positive integers contain at least a term of the current sequence.

Examples

			a(1) = 6 since 6 = 2^2 + 2 with 2 and 2*2+3 = 7 both prime.
a(2) = 7 since 7 = 2^2 + 3 with 2, 3, 2*2+3 all prime.
		

Links

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

Crossrefs

Cf. A000040, A023204, A258139, A258140, A258141.

Programs

Mathematica

n=0;Do[Do[If[PrimeQ[2Prime[k]+3]&&PrimeQ[m-Prime[k]^2],n=n+1;Print[n," ",m];Goto[aa]],{k,1,PrimePi[Sqrt[m]]}];
Label[aa];Continue,{m,1,141}]
Module[{pp=40},Select[Union[#[[1]]^2+#[[2]]&/@Select[Tuples[ Prime[ Range[ pp]],2],PrimeQ[2#[[1]]+3]&]],#<=Prime[pp]-4&]] (* Harvey P. Dale, Jul 24 2021 *)