A091626 Number of ordered integer pairs (b,c) with 0 <= b, c <= n such that both roots of x^2+bx+c=0 are integers.

1, 2, 4, 6, 9, 11, 14, 16, 19, 22, 25, 27, 31, 33, 36, 39, 43, 45, 49, 51, 55, 58, 61, 63, 68, 71, 74, 77, 81, 83, 88, 90, 94, 97, 100, 103, 109, 111, 114, 117, 122, 124, 129, 131, 135, 139, 142, 144, 150, 153, 157, 160, 164, 166, 171, 174, 179, 182, 185, 187

Offset

0,2

Comments

Also number of ordered pairs of nonnegative integers (i, j) such that i+j <= n and i*j <= n. - Seiichi Manyama, Sep 04 2021

Links

Griffin N. Macris, Table of n, a(n) for n = 0..9999

Eric Weisstein's World of Mathematics, Quadratic Equation

Formula

a(n) = a(n-1) + ceiling(tau(n)/2) + 1, n>1. - Vladeta Jovovic, Jun 12 2004

a(n) = n + floor(sqrt(n))/2 + A006218(n)/2, n>0. - Griffin N. Macris, Jun 14 2016

Example

The six quadratics for a(3)=6 and their roots are as follows:
x^2 + 0*x + 0; x=0.
x^2 + 1*x + 0; x=0, x=-1.
x^2 + 2*x + 0; x=0, x=-2.
x^2 + 2*x + 1; x=-1.
x^2 + 3*x + 0; x=0, x=-3.
x^2 + 3*x + 2; x=-1, x=-2.

Mathematica

a[n_] := a[n] = a[n-1] + Ceiling[ DivisorSigma[0, n]/2] + 1; a[0]=1; a[1]=2; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Nov 08 2012, after Vladeta Jovovic *)

Prog

(PARI) a(n) = sum(i=0, n, sum(j=i, n-i, i*j<=n)); \\ Seiichi Manyama, Sep 04 2021
(Python)
from math import isqrt
def A091626(n):
    m = isqrt(n)
    return 1 if n == 0 else n+sum(n//k for k in range(1, m+1))-m*(m-1)//2 # Chai Wah Wu, Oct 07 2021

Crossrefs

Cf. A000005, A006218, A038548, A067274, A091627.

Sequence in context: A265286 A186343 A224995 * A085223 A163057 A242137

Adjacent sequences: A091623 A091624 A091625 * A091627 A091628 A091629

Keyword

nonn

Author

Eric W. Weisstein, Jan 24 2004

Status

approved