A067274 Number of ordered integer pairs (b,c), with -n<=b<=n, -n<=c<=n, such that both roots of x^2+bx+c=0 are integers.

1, 4, 10, 16, 25, 31, 41, 47, 57, 66, 76, 82, 96, 102, 112, 122, 135, 141, 155, 161, 175, 185, 195, 201, 219, 228, 238, 248, 262, 268, 286, 292, 306, 316, 326, 336, 357, 363, 373, 383, 401, 407, 425, 431, 445, 459, 469, 475, 497, 506, 520, 530, 544, 550, 568

Offset

0,2

Comments

Conjecture: The difference a(n)-a(n-1) is 6 if and only if n is a prime number. This has been checked up to about n=300 and may be easy to prove.

Preceding conjecture is a corollary of Jovovic's formula below. - Eric M. Schmidt, Aug 19 2012

Links

Eric M. Schmidt, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Quadratic Equation

Formula

a(n) = a(n-1)+2*(tau(n)+1)+A010052(n), n>1, a(1) = 4. - Vladeta Jovovic, Mar 05 2002, edited by Eric M. Schmidt, Aug 19 2012

Mathematica

a[n_] := If[n >= 1, 2 Sum[Length[Divisors[k]], {k, n}] + Floor[Sqrt[n]] + 2 n - 1]; Join[{1}, Table[a[n], {n, 50}]] (* Lorenz H. Menke, Jr., Apr 13 2016 *)

Prog

(Sage)
def A067274(max) :
    res = [1]
    term = 4
    for ii in range(1, max+1) :
        res += [term]
        term += 2 * (number_of_divisors(ii+1) + 1) + is_square(ii+1)
    return res
# Eric M. Schmidt, Aug 19 2012

Crossrefs

Cf. A010052.

Sequence in context: A271911 A322948 A277368 * A331081 A054901 A019574

Adjacent sequences: A067271 A067272 A067273 * A067275 A067276 A067277

Keyword

nonn

Author

John W. Layman, Feb 21 2002

Status

approved