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

%I A068310 #38 Aug 19 2024 02:04:46
%S A068310 3,2,15,6,35,3,7,5,11,30,143,42,195,14,255,2,323,10,399,110,483,33,23,
%T A068310 39,3,182,87,210,899,15,1023,17,1155,34,1295,38,1443,95,1599,105,1763,
%U A068310 462,215,506,235,138,47,6,51,26,2703,78,2915,21,3135,203,3363,870,3599
%N A068310 n^2 - 1 divided by its largest square divisor.
%C A068310 In other words, squarefree part of n^2-1.
%C A068310 Least m for which x^2 - m*y^2 = 1 has a solution with x = n.
%H A068310 Reinhard Zumkeller, <a href="/A068310/b068310.txt">Table of n, a(n) for n = 2..10000</a>
%F A068310 a(n) = A007913(n^2-1).
%F A068310 a(n) = A005563(n-1) / A008833(n^2 - 1). - _Reinhard Zumkeller_, Nov 26 2011; corrected by _Georg Fischer_, Dec 10 2022
%e A068310 a(6) = 35, as 6^2 - 1 = 35 itself is squarefree.
%e A068310 7^2-1 = 48 = A005563(6), whose largest square divisor is A008833(48) = 16, so a(7) = 48/16 = 3.
%t A068310 a[n_] := Times@@(#[[1]] ^ Mod[ #[[2]], 2]&/@FactorInteger[n^2-1])
%t A068310 Table[(n^2-1)/Max[Select[Divisors[n^2-1],IntegerQ[Sqrt[#]]&]],{n,2,60}] (* _Harvey P. Dale_, Dec 08 2019 *)
%o A068310 (PARI) a(n) = core(n*n - 1); \\ _David Wasserman_, Mar 07 2005
%o A068310 (Haskell)
%o A068310 a068310 n = f 1 $ a027746_row (n^2 - 1) where
%o A068310    f y [] = y
%o A068310    f y [p] = y*p
%o A068310    f y (p:ps'@(p':ps)) | p == p' = f y ps
%o A068310                        | otherwise = f (y*p) ps'
%o A068310 -- _Reinhard Zumkeller_, Nov 26 2011
%Y A068310 Cf. A002350, A007913, A067872, A033314, A027746, A175607.
%K A068310 easy,nice,nonn
%O A068310 2,1
%A A068310 _Lekraj Beedassy_, Feb 25 2002
%E A068310 Edited by _Dean Hickerson_, Mar 19 2002
%E A068310 Entry revised by _N. J. A. Sloane_, Apr 27 2007