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

Author

Lekraj Beedassy, Feb 25 2002

Keywords

• nice
• nonn

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

Considering the greatest prime factor (gpf) of terms, two curves are evident. One has gpf ~ n^2 (see a(9949)), the other has gpf ~ n^2/7 (see a(9923)). See the gpf plot link. - Bill McEachen, Apr 13 2025

Examples

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

Links

• T. D. Noe and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n = 1..500 from T. D. Noe
• Bill McEachen, Plot of gpf(sequence terms)

Crossrefs

Cf. A033318, A068310.

Cf. A010052.

Programs

Haskell

a067872 n = (until ((== 1) . a010052 . (+ 1)) (+ nn) nn) `div` nn
            where nn = n ^ 2
-- Reinhard Zumkeller, Jun 28 2013

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 *)

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

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

Extensions

Edited by Dean Hickerson, Mar 19 2002