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

1,2

Comments

Conjecture: The current sequence has infinitely many terms.

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

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.)

Example

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.

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

Crossrefs

Cf. A000040, A000290, A232174.

Sequence in context: A055699 A244378 A062094 * A200543 A363453 A246426

Adjacent sequences: A334835 A334836 A334837 * A334839 A334840 A334841

Keyword

nonn

Author

Zhi-Wei Sun, May 13 2020

Status

approved