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

0,4

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

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000005, A067274, A091626, A094820.

Sequence in context: A014248 A174799 A248566 * A160830 A363761 A363763

Adjacent sequences: A091624 A091625 A091626 * A091628 A091629 A091630

Keyword

nonn

Author

Eric W. Weisstein, Jan 24 2004

Status

approved