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A269254 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. 16
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

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 Fanatics Discussion 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[1] = a[2] = 1;

a[n_ /; IntegerQ[Sqrt[n+2]]] = -1;

a[n_] := Module[{s}, s[0] = 1; s[1] = 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-Fran├žois 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: A072047 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 February 21 02:57 EST 2019. Contains 320364 sequences. (Running on oeis4.)