A084702 a(n) is the smallest k such that k + 1 and n*k + 1 both are perfect squares, or 0 if no such number exists.

3, 24, 8, 0, 3, 8, 24, 3, 0, 8, 48, 24, 15, 120, 8, 3, 15, 168, 80, 48, 3, 24, 360, 15, 0, 24, 440, 8, 120, 80, 120, 195, 3, 840, 24, 8, 35, 960, 440, 3, 168, 120, 168, 28560, 8, 48, 1680, 35, 0, 48, 24, 120, 483, 175560, 8, 3, 24, 528, 212520, 728, 63, 3024

Offset: 1

Author

Amarnath Murthy, Jun 08 2003

Keywords

• nonn

Comments

a(4) = 0 as when k+1 is a square, 4k+4 is also a square; hence 4k+1 cannot be a square for k > 0.

Examples

			a(5) = 3 as 3 + 1 = 4 and 3*5 + 1 = 16 both are squares.
		

Links

• David Wasserman, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A084703.

Programs

Mathematica

r[n_, c_] := Reduce[i>1 && j>1 && k+1 == i^2 && n*k+1 == j^2, {i,j,k}, Integers] /. C[1] -> c // Simplify;
a[n_] := If[rn = r[n,0] || r[n,1] || r[n,2]; rn === False, 0, k /. Solve[rn] // Min];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 100}] (* Jean-François Alcover, May 12 2017 *)

Formula

a(i^2-1) is usually (i-1)^2-1. For 2 < i < 1000 there are 34 exceptions. The first four of these are a(11^2-1) = 3, a(23^2-1) = 8, a(39^2-1) = 15 and a(41^2-1) = 3. - David Wasserman, May 03 2007

Extensions

More terms from Donald Sampson (marsquo(AT)hotmail.com), Dec 04 2003

Corrected by David Wasserman, May 03 2007