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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.
10
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 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
KEYWORD
nonn
AUTHOR
John W. Layman, Feb 21 2002
STATUS
approved