A391597 a(n) is the number of quadratic polynomials with coefficients in {-n, ..., n}, positive leading coefficient, and having two integer roots.

4, 14, 24, 43, 53, 79, 89, 118, 137, 163, 173, 222, 232, 258, 284, 326, 336, 385, 395, 444, 470, 496, 506, 583, 602, 628, 657, 706, 716, 786, 796, 852, 878, 904, 930, 1023, 1033, 1059, 1085, 1162, 1172, 1242, 1252, 1301, 1350, 1376, 1386, 1498, 1517, 1566, 1592, 1641, 1651, 1728

Offset

1,1

Links

Table of n, a(n) for n=1..54.

Formula

a(n) = Sum_{u = 1..n} b(floor(n/u)), where b = A067274 (obtained in collaboration with Sean A. Irvine). Proof sketch: consider a polynomial u*x^2+v*x+w with integer roots r and s. Then v = -u*(r+s) and w = u*r*s; so the constraint max(|v|, |w|) <= n is equivalent to max(|r+s|, |r*s|) <= floor(n/u). Therefore, the number of admissible pairs (r, s) is b(floor(n/u)) and the formula follows.

a(n) = 2*(A006218(n)+A061201(n))+A013936(n)-n. - Chai Wah Wu, Jan 28 2026

Example

For n = 1, the a(1) = 4 polynomials with coefficients in {-1, 0, 1} are: x^2-x, x^2-1, x^2, x^2+x.
For n = 2, the a(2) = 14 polynomials with coefficients in {-2, -1, 0, 1, 2} are: x^2-2x, x^2-2x+1, x^2-x-2, x^2-x, x^2-1, x^2, x^2+x-2, x^2+x, x^2+2x, x^2+2x+1, 2x^2-2x, 2x^2-2, 2x^2, 2x^2+2x.

Prog

(Python)
from math import isqrt
def A067274(n): return (n-(s:=isqrt(n))**2+(sum(n//k for k in range(1, s+1))<<1)<<1)+s-1 if n else 1
def A391597(n):
    c, j = 0, 1
    while j <= n:
        c += A067274(k:=n//j)*(-j+(j:=n//k+1))
    return c # Chai Wah Wu, Jan 28 2026

Crossrefs

Cf. A067274, A391527 (quadratic, rational), A391108 (cubic, integer), A392944 (cubic, rational), A006218, A061201, A013936.

Sequence in context: A277591 A017317 A195973 * A116963 A094930 A289665

Adjacent sequences: A391594 A391595 A391596 * A391598 A391599 A391600

Keyword

nonn

Author

Lorenzo Sauras Altuzarra, Jan 12 2026

Status

approved