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 A269254 To find a(n), define a sequence by s(k) = n*s(k-1) - s(k-2), with s(0) = 1, s(1) = n + 1; then a(n) is the smallest index k such that s(k) is prime, or -1 if no such k exists. 17
 1, 1, 2, 1, 2, 1, -1, 2, 2, 1, 2, 1, 2, -1, 2, 1, 3, 1, 2, 2, 2, 1, -1, 2, 6, 2, 3, 1, 3, 1, 2, 9, 9, -1, 2, 1, 6, 2, 2, 1, 2, 1, 5, 2, 2, 1, -1, 2, 5, 2, 9, 1, 2, 2, 2, 2, 6, 1, 2, 1, 14, -1, 5, 2, 2, 1, 5, 2, 3, 1, 6, 1, 8, 3, 6, 2, 3, 1, -1, 3, 18, 1, 2, 3, 2, 2, 3, 1, 2, 9, 3, 5, 2, 2, 96, 1, 3, -1, 5, 1, 2, 1, 2, 15, 14, 1, 44, 1, 3, -1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The s(k) sequences can be viewed in A294099, where they appear as rows. - Peter Munn, Aug 31 2020 For n >= 3, a(n) is that 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. Every positive term belongs to A005097. When n=7, the sequence {s(k)} is A033890, which is Fibonacci(4i+2), and since x|y <=> F_x|F_y, and 2i+1|4i+2, A033890 is never prime, and so a(7)=-1. For the other -1 terms below 100, see the theorem below and the Klee link - N. J. A. Sloane, Oct 20 2017 and Oct 22 2017 Theorem (Brad Klee): For all n > 2, a(n^2 - 2) = -1. See Klee link for a proof. - L. Edson Jeffery, Oct 22 2017 Theorem (Based on work of Hans Havermann, L. Edson Jeffery, Brad Klee, Don Reble, Bob Selcoe, and N. J. A. Sloane) a(110) = -1. [For proof see link. - N. J. A. Sloane, Oct 23 2017] From Bob Selcoe, Oct 24 2017, edited by N. J. A. Sloane, Oct 27 2017: (Start) Suppose n = m^2 - 2, where m >= 3, and let j = m-2, with j >= 1. For this value of n, the sequence s(k) satisfies s(k) = (c(k) + d(k))*(c(k) - d(k)), where c(0) = 1, d(0) = 0; and for k >= 1: c(k) = (j+2)*c(k-1) - d(k-1), and d(k) = c(k-1). So (as Brad Klee already proved) a(n) = -1 . We have s(0) = 1 and s(1) = n+1 = j^2 + 4j + 3. In general, the coefficients of s(k) when expanded in powers of j are given by the (4k+2)-th row of A011973 (the triangle of coefficients of Fibonacci polynomials) in reverse order. For example, s(2) = j^4 + 8j^3 + 21j^2 + 20j + 5, s(3) = j^6 + 12j^5 + 55j^4 + 120j^3 + 126j^2 + 56j + 7, etc. Perhaps the above comments could be generalized to apply to a(110) or to other n for which a(n) = -1? (End) For detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018 LINKS Hans Havermann, Table of n, a(n) for n = 1..946 Hans Havermann, Table of n, a(n) for n = 1..10000 C. K. Caldwell, Top Twenty page, Lehmer number Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018. Brad Klee, Proof for A269254, Sequence Fans Mailing List, October 2017. N. J. A. Sloane et al., Proof that a(110) = -1 Wikipedia, Lehmer number. FORMULA If n is prime then a(n-1) = 1. EXAMPLE Let b(k) be the recursive sequence defined by the initial conditions b(0) = 1, b(1) = 16, and the recursive equation b(k) = 15*b(k-1) - b(k-2). a(15) = 2 because b(2) = 239 is the smallest prime in b(k). Let c(k) be the recursive sequence defined by the initial conditions c(0) = 1, c(1) = 18, and the recursive equation c(k) = 17*c(k-1) - c(k-2). a(17) = 3 because c(3) = 5167 is the smallest prime in c(k). MATHEMATICA kmax = 100; a = a = 1; a[n_ /; IntegerQ[Sqrt[n+2]]] = -1; a[n_] := Module[{s}, s = 1; s = n+1; s[k_] := s[k] = n s[k-1] - s[k-2]; For[k=1, k <= kmax, k++, If[PrimeQ[s[k]], Return[k]]]; Print["For n = ", n, ", k = ", k, " exceeds the limit kmax = ", kmax]; -1]; Array[a, 110] (* Jean-François Alcover, Aug 05 2018 *) PROG (MAGMA) lst:=[]; for n in [1..85] do if n gt 2 and IsSquare(n+2) then Append(~lst, -1); else a:=n+1; c:=1; t:=1; if IsPrime(a) then Append(~lst, t); else repeat b:=n*a-c; c:=a; a:=b; t+:=1; until IsPrime(a); Append(~lst, t); end if; end if; end for; lst; (PARI) allocatemem(2^30); default(primelimit, (2^31)+(2^30)); s(n, k) = if(0==k, 1, if(1==k, (1+n), ((n*s(n, k-1)) - s(n, k-2)))); A269254(n) = { my(k=1); if((n>2)&&issquare(2+n), -1, while(!isprime(s(n, k)), k++); (k)); }; \\ Antti Karttunen, Oct 20 2017 CROSSREFS Cf. A005097, A011973, A269251, A269252, A269253. Cf. A294099 (array used to compute this sequence). Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A298675, A298677, A298878, A299045, A299071. Sequence in context: A327521 A282870 A106802 * A049236 A244259 A094840 Adjacent sequences:  A269251 A269252 A269253 * A269255 A269256 A269257 KEYWORD sign AUTHOR Arkadiusz Wesolowski, Jul 09 2016 EXTENSIONS a(86)-a(94) from Antti Karttunen, Oct 20 2017 a(95)-a(109) appended by L. Edson Jeffery, Oct 22 2017 STATUS approved

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Last modified June 24 17:51 EDT 2021. Contains 345419 sequences. (Running on oeis4.)