A084866 Primes that can be written in the form 2*p^2 + 3*q^2 with p and q prime.

83, 173, 197, 269, 317, 389, 461, 557, 653, 701, 797, 941, 1091, 1109, 1181, 1229, 1637, 1709, 1949, 1997, 2069, 2141, 2309, 2531, 2549, 2621, 2789, 2861, 3221, 3389, 3461, 3581, 3821, 4157, 4229, 4349, 4493, 5051, 5261, 5381, 5501, 5693

Offset: 1

Author

Reinhard Zumkeller, Jun 10 2003

Keywords

• nonn

Comments

Subsequence of A084864 and of A084865; A084863(a(n))>0.

Examples

			A000040(40) = 173 = 98 + 75 = 2*7^2 + 3*5^2 = 2*A000040(4)^2 + 3*A000040(3)^2, therefore 173 is a term.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A084863, A084864, A084865.

Programs

Maple

N:= 10^4: # to get terms <= N
P:= select(isprime, [2,seq(i,i=3..floor((N/2)^(1/2)))]):
m:= nops(P):
R:= {}:
for p in P do
  for i from 2 to m while 3*P[i]^2 <= N - 2*p^2 do
    v:= 2*p^2 + 3*P[i]^2;
    if isprime(v) then R:= R union {v} fi
od od:
sort(convert(R,list)); # Robert Israel, Nov 05 2020

Mathematica

nn = 10^4; (* to get terms <= nn *)
P = Select[Join[{2}, Range[3, Floor[Sqrt[nn/2]]]], PrimeQ];
m = Length[P];
R = {};
Do[For[i = 2, 3*P[[i]]^2 <= nn - 2*p^2, i++,
     v = 2*p^2 + 3*P[[i]]^2;
     If[PrimeQ[v], R = R ~Union~ {v}]],
{p, P}];
Sort[R] (* Jean-François Alcover, Dec 13 2021, after Robert Israel *)