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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified May 20 09:27 EDT 2022. Contains 353871 sequences. (Running on oeis4.)