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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. 9
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 (list; graph; refs; listen; history; text; internal format)
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
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
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

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)