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

Author

Eric W. Weisstein, Jan 24 2004

Keywords

• nonn

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

Examples

			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.
		

Links

• Griffin N. Macris, Table of n, a(n) for n = 0..9999
• Eric Weisstein's World of Mathematics, Quadratic Equation

Crossrefs

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

Programs

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 *)

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

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