A334838 Positive integers m with prime(m) in the form x^2 + m*y^2, where x and y are positive integers.

1, 2, 12, 35, 37, 77, 97, 100, 118, 136, 137, 152, 183, 184, 190, 212, 231, 258, 290, 352, 421, 462, 482, 487, 690, 730, 741, 960, 1110, 1111, 1168, 1169, 1227, 1285, 1328, 1396, 1417, 1621, 2074, 2119, 2318, 2578, 2603, 2652, 2707, 2726, 2737, 2772, 2776, 2788, 2803, 2853, 2857, 2865, 2882, 2892, 3035, 3176, 3199, 3245

Offset: 1

Author

Zhi-Wei Sun, May 13 2020

Keywords

• nonn

Comments

Conjecture: The current sequence has infinitely many terms.

This was first mentioned in Remark 2.21 of the linked 2017 paper.

Examples

			a(2) = 2 with prime(2) = 3 = 1^2 + 2*1^2.
a(3) = 12 with prime(12) = 37 = 5^2 + 12*1^2.
a(4) = 35 with prime(35) = 149 = 3^2 + 35*2^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv, arXiv:1211.1588 [math.NT], 2012-2017.)

Crossrefs

Cf. A000040, A000290, A232174.

Programs

Mathematica

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
tab={};Do[Do[If[SQ[Prime[m]-m*x^2],tab=Append[tab,m];Goto[aa]],{x,1,Sqrt[Prime[m]/m]}];Label[aa],{m,1,3245}];tab