A334838 - OEIS

%I #6 May 15 2020 13:08:26

%S 1,2,12,35,37,77,97,100,118,136,137,152,183,184,190,212,231,258,290,

%T 352,421,462,482,487,690,730,741,960,1110,1111,1168,1169,1227,1285,

%U 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

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

%C Conjecture: The current sequence has infinitely many terms.

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

%H Zhi-Wei Sun, <a href="/A334838/b334838.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, 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 <a href="http://arxiv.org/abs/1211.1588">arXiv</a>, arXiv:1211.1588 [math.NT], 2012-2017.)

%e a(2) = 2 with prime(2) = 3 = 1^2 + 2*1^2.

%e a(3) = 12 with prime(12) = 37 = 5^2 + 12*1^2.

%e a(4) = 35 with prime(35) = 149 = 3^2 + 35*2^2.

%t SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];

%t 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

%Y Cf. A000040, A000290, A232174.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, May 13 2020