A384666 Number of distinct values of the quadratic discriminant D=b^2-4*a*c, for a,b,c in the range [-n,n].

1, 6, 17, 35, 56, 90, 125, 178, 223, 282, 344, 436, 499, 608, 701, 804, 904, 1062, 1164, 1339, 1450, 1604, 1765, 1988, 2114, 2335, 2525, 2735, 2909, 3194, 3366, 3679, 3887, 4137, 4389, 4661, 4840, 5237, 5536, 5835, 6068, 6507, 6759, 7195, 7473, 7773, 8148, 8645

Offset: 0

Author

Darío Clavijo, Jun 06 2025

Keywords

• nonn

Comments

a(n) is lower bounded by n log n for n > 0.

The number of distinct a*c is 2*A027384(n)-1.

Crossrefs

Cf. A000217, A027384.

Programs

Python

def a(n):
    D, ac = {0}, {0}
    SQ = [i*i for i in range(0, n+1)]
    for i in range(1, n+1):
        ac.add(i)
        if (s:= SQ[i]) > n:
            ac.add(s)
    for a_ in range(2, n):
        for c in range(a_+ 1, n+1):
            ac.add(a_* c)
    for b in range(n + 1):
        b2 = SQ[b]
        for v in ac:
            ac4 = v << 2
            D.add(b2 + ac4)
            if b2 < ac4:
                D.add(b2 - ac4)
    return len(D)
print([a(n) for n in range(0, 48)])