Also primes of the forms x^2 + 48*y^2 and x^2 + 72*y^2. See A140633. - T. D. Noe, May 19 2008
Primes of the quadratic form are a subset of the primes congruent to 1 (mod 24). [Proof. For 0 <= x, y <= 23, the only values mod 24 that x^2 + 24*y^2 can take are 0, 1, 4, 9, 12 or 16. All of these r except 1 have gcd(r, 24) > 1 so if x^2 + 24*y^2 is prime its remainder mod 24 must be 1.] - David A. Corneth, Jun 08 2020
More advanced mathematics seems to be needed to determine whether this sequence lists all primes congruent to 1 (mod 24). Note the significance of 24 being a convenient number, as described in A000926. See also Sloane et al., Binary Quadratic Forms and OEIS, which explains how the table in A139642 may be used for this determination. - Peter Munn, Jun 21 2020
Primes == 1 (mod 2^3*3) are the intersection of the primes == 1 (mod 2^3) in A007519 and the primes == 1 (mod 3) in A002476, by the Chinese remainder theorem. - R. J. Mathar, Jun 11 2020
This is indeed the same as primes p == 1 (mod 24) since the discriminant -96 is a term in A003171 (i.e., there is 1 class of binary quadratic forms per genus for discriminant -96), so the primes represented by any binary quadratic form are determined by a congruence condition.
Also primes of the form x^2 + 8*x*y - 8*y^2 (as well as of the form x^2 + 10*x*y + y^2, or x^2 - 24*y^2), since the discriminant (+96) of these forms is a term in A390079 (also 1 class per genus), and it happens that the primes represented by these forms are given by the same congruence condition p == 1 (mod 24). (End)
REFERENCES
Z. I. Borevich and I. R. Shafarevich. Number Theory. Academic Press. 1966.
LINKS
Vincenzo Librandi, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi, next 143 terms from N. J. A. Sloane]
