A067872 Least m > 0 for which m*n^2 + 1 is a square.

3, 2, 7, 3, 23, 8, 47, 15, 79, 24, 119, 2, 167, 48, 3, 63, 287, 80, 359, 6, 88, 120, 527, 28, 623, 168, 727, 12, 839, 44, 959, 255, 216, 288, 8, 20, 1367, 360, 19, 77, 1679, 22, 1847, 30, 208, 528, 2207, 7, 2399, 624, 128, 42, 2807, 728, 696, 3, 160, 840, 3479, 11

Offset

1,1

Comments

Least m > 0 for which x^2 - m*y^2 = 1 has a solution with y = n.

For n > 1, a(n) <= n^2-2. - Chai Wah Wu, Jan 26 2016

Links

T. D. Noe and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..500 from T. D. Noe

Formula

For n a power of an odd prime, a(n) = n^2 - 2. For n twice a power of an odd prime, a(n) = (n/2)^2 - 1. - T. D. Noe, Sep 13 2007

Example

a(4)=3, based on 3*4^2 + 1 = 7^2.

Mathematica

a[n_] := For[m=1, True, m++, If[IntegerQ[Sqrt[m*n^2+1]], Return[m]]]; Table[a[n], {n, 100}]
lm[n_]:=Module[{m=1}, While[!IntegerQ[Sqrt[m n^2+1]], m++]; m]; Array[lm, 60] (* Harvey P. Dale, Feb 24 2013 *)

Prog

(Haskell)
a067872 n = (until ((== 1) . a010052 . (+ 1)) (+ nn) nn) `div` nn
            where nn = n ^ 2
-- Reinhard Zumkeller, Jun 28 2013
(Python)
def A067872(n):
    y, x, n2 = n*(n+2), 2*n+3, n**2
    m, r = divmod(y, n2)
    while r:
        y += x
        x += 2
        m, r = divmod(y, n2)
    return m # Chai Wah Wu, Jan 25 2016

Crossrefs

Cf. A033318, A068310.

Cf. A010052.

Sequence in context: A241559 A165794 A075270 * A318457 A198501 A230072

Adjacent sequences: A067869 A067870 A067871 * A067873 A067874 A067875

Keyword

nice,nonn

Author

Lekraj Beedassy, Feb 25 2002

Extensions

Edited by Dean Hickerson, Mar 19 2002

Status

approved