A269252 - OEIS

%I #50 Apr 03 2023 10:36:13

%S -1,-1,1,1,2,1,2,1,2,2,2,1,3,1,3,2,2,1,14,1,2,2,3,1,2,5,2,36,2,1,2,1,

%T 15,-1,6,2,3,1,2,2,6,1,3,1,2,2,2,1,2,3,2,-1,3,1,2,2,2,6,3,1,2,1,30,3,

%U 2,2,2,1,2,5,2,1,5,1,6,3,2,6,3,1,8,6,14,1,3

%N Define a sequence by s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n - 1. a(n) is the smallest index k such that s(k) is prime, or -1 if no such k exists.

%C For n >= 2, positive integer k yielding the smallest prime of the form (x^y + 1/x^y)/(x + 1/x), where x = (sqrt(n+2) +/- sqrt(n-2))/2 and y = 2*k + 1, or -1 if no such k exists.

%C Every positive term belongs to A005097.

%C For detailed theory, see [Hone]. - _L. Edson Jeffery_, Feb 09 2018

%H C. K. Caldwell, Top Twenty page, <a href="https://t5k.org/top20/page.php?id=47">Lehmer number</a>

%H Andrew N. W. Hone, et al., <a href="https://arxiv.org/abs/1802.01793">On a family of sequences related to Chebyshev polynomials</a>, arXiv:1802.01793 [math.NT], 2018.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lehmer_number">Lehmer number</a>

%F If n is prime then a(n+1) = 1.

%e Let b(k) be the recursive sequence defined by the initial conditions b(0) = 1, b(1) = 10, and the recursive equation b(k) = 11*b(k-1) - b(k-2). a(11) = 2 because b(2) = 109 is the smallest prime in b(k).

%e Let c(k) be the recursive sequence defined by the initial conditions c(0) = 1, c(1) = 12, and the recursive equation c(k) = 13*c(k-1) - c(k-2). a(13) = 3 because c(3) = 2003 is the smallest prime in c(k).

%t s[k_, m_] := s[k, m] = Which[k == 0, 1, k == 1, 1 + m, True, m s[k - 1, m] - s[k - 2, m]]; Table[SelectFirst[Range[120], PrimeQ@ Abs@ s[#, -n] &] /. k_ /; MissingQ@ k -> -1, {n, 85}] (* _Michael De Vlieger_, Feb 03 2018 *)

%o lst:=[]; for n in [1..85] do if n in [1, 2, 34, 52] then Append(~lst, -1); else a:=1; c:=1; t:=0; repeat b:=n*a-c; c:=a; a:=b; t+:=1; until IsPrime(a); Append(~lst, t); end if; end for; lst;

%Y Cf. A005097, A269251, A269253, A269254, A299045 (array used to compute this sequence).

%Y Cf. A294099, A298675, A298677, A298878, A299071, A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501.

%K sign

%O 1,5

%A _Arkadiusz Wesolowski_, Jul 09 2016