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A130280 a(n) = smallest integer k>1 such that n(k^2-1)+1 is a perfect square, or 0 if no such number exists.

%I A130280 #15 May 23 2019 08:30:10
%S A130280 2,5,3,0,2,3,5,2,0,3,7,5,4,11,3,2,4,13,9,7,2,5,19,4,0,5,21,3,11,9,11,
%T A130280 14,2,29,5,3,6,31,21,2,13,11,13,169,3,7,41,6,0,7,5,11,22,419,3,2,5,23,
%U A130280 461,27,8,55,7,4,2,3,49,29
%N A130280 a(n) = smallest integer k>1 such that n(k^2-1)+1 is a perfect square, or 0 if no such number exists.
%C A130280 A084702(n) = a(n)^2-1, resp. a(n) = sqrt(A084702(n)+1). See A130283 for values where A130280(n)=0.
%H A130280 M. F. Hasler, <a href="/A130280/b130280.txt">Table of n, a(n) for n = 1..1000</a>
%F A130280 If n=(2k)^2, then A130280(n) <= k, since (2k)^2(k^2-1)+1 = (2k^2-1)^2. See A130281 for the cases where equality does not hold. If n=k^2-1, then A130280(n) <= k-1 since (k^2-1)((k-1)^2-1)+1 = (k^2-k-1)^2. See A130282 for the cases where equality does not hold.
%e A130280 a( (2k)^2 ) <= k since (2k)^2(k^2-1)+1 = (2k^2-1)^2 (but k=1 is excluded since with k^2-1=0 this would be a trivial solution for any n).
%p A130280 A130280:=proc(n) local x,y,z; if n=1 then return 2 fi; isolve(n*(x^2-1)+1=y^2,z); select(has,`union`(%),x); map(rhs,%); simplify(eval(%,z=1) union eval(%,z=0)) minus {-1,1}; if %={} then 0 else (min@op@map)(abs,%) fi end;
%t A130280 $MaxExtraPrecision = 100;
%t A130280 r[n_, c_] := Reduce[k > 1 && j > 1 && n*(k^2 - 1) + 1 == j^2, {j, k}, Integers] /. C[1] -> c // Simplify;
%t A130280 a[n_] := If[rn = r[n,0] || r[n,1] || r[n,2]; rn === False, 0, k /. {ToRules[rn]} // Min];
%t A130280 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 800}] (* _Jean-François Alcover_, May 12 2017 *)
%o A130280 (PARI) {A130280(n,L=10^15)=if(issquare(n),L=2+sqrtint(n>>2)); for( k=2, L, if( issquare(n*(k^2-1)+1),return(k)))}
%Y A130280 Cf. A084702, A094357, A130281, A130282, A130283, A130284.
%Y A130280 See also A306767.
%K A130280 easy,nonn
%O A130280 1,1
%A A130280 _M. F. Hasler_, May 20 2007, May 25 2007