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

1,1

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.

Links

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

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

Example

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

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 *)

Crossrefs

Cf. A084703.

Sequence in context: A120085 A062834 A047980 * A367190 A168061 A261381

Adjacent sequences: A084699 A084700 A084701 * A084703 A084704 A084705

Keyword

nonn

Author

Amarnath Murthy, Jun 08 2003

Extensions

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

Corrected by David Wasserman, May 03 2007

Status

approved