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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.
9
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
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
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jan 24 2004
STATUS
approved