A068310 n^2 - 1 divided by its largest square divisor.

3, 2, 15, 6, 35, 3, 7, 5, 11, 30, 143, 42, 195, 14, 255, 2, 323, 10, 399, 110, 483, 33, 23, 39, 3, 182, 87, 210, 899, 15, 1023, 17, 1155, 34, 1295, 38, 1443, 95, 1599, 105, 1763, 462, 215, 506, 235, 138, 47, 6, 51, 26, 2703, 78, 2915, 21, 3135, 203, 3363, 870, 3599

Offset: 2

Author

Lekraj Beedassy, Feb 25 2002

Keywords

• easy
• nice
• nonn

Comments

In other words, squarefree part of n^2-1.

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

Examples

			a(6) = 35, as 6^2 - 1 = 35 itself is squarefree.
7^2-1 = 48 = A005563(6), whose largest square divisor is A008833(48) = 16, so a(7) = 48/16 = 3.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A002350, A007913, A067872, A033314, A027746, A175607.

Programs

Haskell

a068310 n = f 1 $ a027746_row (n^2 - 1) where
   f y [] = y
   f y [p] = y*p
   f y (p:ps'@(p':ps)) | p == p' = f y ps
                       | otherwise = f (y*p) ps'
-- Reinhard Zumkeller, Nov 26 2011

Mathematica

a[n_] := Times@@(#[[1]] ^ Mod[ #[[2]], 2]&/@FactorInteger[n^2-1])
Table[(n^2-1)/Max[Select[Divisors[n^2-1],IntegerQ[Sqrt[#]]&]],{n,2,60}] (* Harvey P. Dale, Dec 08 2019 *)

PARI

a(n) = core(n*n - 1); \\ David Wasserman, Mar 07 2005

Formula

a(n) = A007913(n^2-1).

a(n) = A005563(n-1) / A008833(n^2 - 1). - Reinhard Zumkeller, Nov 26 2011; corrected by Georg Fischer, Dec 10 2022

Extensions

Edited by Dean Hickerson, Mar 19 2002

Entry revised by N. J. A. Sloane, Apr 27 2007