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Conjecture: if D is a term of this sequence, then D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4. For example, a(1) = 12 = 3*4, a(2) = 21 = 3*7, a(3) = 24 = 3*8, a(4) = 28 = 4*7, a(5) = 33 = 3*11, ... [This conjecture is correct: see Theorem 1 and Theorem 2 of Ezra Brown link; see also A003656. - Jianing Song, Dec 28 2021]

Let k be the quadratic field with discriminant D, O_k be ring of integers of k, N(x) be the norm of x and (D/p) be the Kronecker symbol. If D is a term of this sequence and D = uv, where u, v are 4, 8 or primes congruent to 3 modulo 4, then:

(a) if (((-u)/p), ((-v)/p)) = (1, 1), (1, 0) or (0, 1), then N(x) = p has solutions in O_k, while N(y) = -p has no solutions in k. For example, for D = 21 and p = 37, we have ((-3)/37) = ((-7)/37) = 1, and N(x) = 37 has solution x = (13 + sqrt(21))/2, but N(y) = -37 has no solutions in Q(sqrt(21)).

(b) if (((-u)/p), ((-v)/p)) = (-1, -1), (-1, 0) or (0, -1), then N(x) = -p has solutions in O_k, while N(y) = p has no solutions in k. For example, for D = 12 and p = 11, we have ((-3)/11) = ((-4)/11) = -1, and N(x) = -11 has solution x = 1 + 2*sqrt(3), but N(y) = 11 has no solutions in Q(sqrt(3)).

(c) if (((-u)/p), ((-v)/p)) = (1, -1) or (-1, 1), then N(x) = +-p has no solutions in k.

The smallest number of the form above that is not in this sequence is 316 = 4*79.

Also, it is conjectured that the quadratic field with discriminant D has form class number 2, where D is a term of this sequence. This is equivalent to the conjecture above. [This can also be deduced from the first paragraph of Ezra Brown link: the norm of the fundamental unit of the field k is -1 if D = 8 or a prime congruent to 1 modulo 4, and 1 if D is in this sequence. Here k is the quadratic field with discriminant D. - Jianing Song, Dec 28 2021]

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.

This is a subsequence of A002144, A002313, A003655, A050150, A062090, A141293, A175768, A192592, A226181 (conjectural).