A091627 Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.

0, 0, 1, 2, 4, 5, 7, 8, 10, 12, 14, 15, 18, 19, 21, 23, 26, 27, 30, 31, 34, 36, 38, 39, 43, 45, 47, 49, 52, 53, 57, 58, 61, 63, 65, 67, 72, 73, 75, 77, 81, 82, 86, 87, 90, 93, 95, 96, 101, 103, 106, 108, 111, 112, 116, 118, 122, 124, 126, 127, 133, 134, 136, 139, 143

Offset: 0

Author

Eric W. Weisstein, Jan 24 2004

Keywords

• nonn

Comments

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Quadratic Equation

Crossrefs

Cf. A000005, A067274, A091626, A094820.

Programs

Mathematica

Accumulate[ Join[{0, 0}, Table[ Ceiling[ DivisorSigma[0, n]/2], {n, 2, 64}]]]  (* Jean-François Alcover, Oct 23 2012, after Vladeta Jovovic *)

PARI

a(n) = sum(i=1, n, sum(j=i, n-i, i*j<=n)); \\ Seiichi Manyama, Sep 04 2021

PARI

N=66; x='x+O('x^N); concat([0, 0], Vec((-x+sum(k=1, sqrtint(N), x^k^2/(1-x^k)))/(1-x))) \\ Seiichi Manyama, Sep 04 2021

Python

from math import isqrt
def A091627(n):
    m = isqrt(n)
    return 0 if n == 0 else sum(n//k for k in range(1, m+1))-m*(m-1)//2-1 # Chai Wah Wu, Oct 07 2021

Formula

a(n) = A091626(n) - n - 1. a(n) = a(n-1) + ceiling(tau(n)/2), n>1. Partial sums of A038548. - Vladeta Jovovic, Jun 12 2004

G.f.: (1/(1 - x)) * (-x + Sum_{k>=1} x^(k^2)/(1 - x^k)). - Seiichi Manyama, Sep 04 2021