

A269253


Smallest prime in the sequence s(k) = n*s(k1)  s(k2), with s(0) = 1, s(1) = n + 1 (or 1 if no such prime exists).


15



2, 3, 11, 5, 29, 7, 1, 71, 89, 11, 131, 13, 181, 1, 239, 17, 5167, 19, 379, 419, 461, 23, 1, 599, 251894449, 701, 20357, 29, 25171, 31, 991, 36002209323169, 47468744103199, 1, 1259, 37, 2625505273, 1481, 1559, 41, 1721, 43, 150103799, 1979, 2069, 47, 1, 2351, 287762399
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OFFSET

1,1


COMMENTS

For n >= 3, smallest prime of the form (x^y  1/x^y)/(x  1/x), where x = (sqrt(n+2) +/ sqrt(n2))/2 and y is an odd positive integer, or 1 if no such prime exists.
When n=7, the sequence {s(k)} is A033890, which is Fibonacci(4i+2), and since xy <=> F_xF_y, and 2i+14i+2, A033890 is never prime, and so a(7) = 1. What is the proof for the other entries that are 1? Answer: See the Comments in A269254.  N. J. A. Sloane, Oct 22 2017
For detailed theory, see [Hone].  L. Edson Jeffery, Feb 09 2018


LINKS

Hans Havermann, Table of n, a(n) for n = 1..300
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.
Wikipedia, Lehmer number


FORMULA

If n is prime then a(n1) = n.


MATHEMATICA

terms = 172;
kmax = 120;
a[n_] := Module[{s, k}, s[k_] := s[k] = n s[k1]  s[k2]; s[0] = 1; s[1] = n+1; For[k = 1, k <= kmax, k++, If[PrimeQ[s[k]], Return[s[k]]]]];
Array[a, terms] /. Null > 1 (* JeanFrançois Alcover, Aug 30 2018 *)


PROG

(MAGMA) lst:=[]; for n in [1..49] do if n gt 2 and IsSquare(n+2) then Append(~lst, 1); else a:=n+1; c:=1; if IsPrime(a) then Append(~lst, a); else repeat b:=n*ac; c:=a; a:=b; until IsPrime(a); Append(~lst, a); end if; end if; end for; lst;


CROSSREFS

Cf. A117522, A269251, A269252, A269254.
Cf. A294099, A298675, A298677, A298878, A299045, A299071, A285992, A299107, A299109, A088165, A299100, A299101, A113501.
Sequence in context: A229607 A137332 A292473 * A084047 A282770 A145077
Adjacent sequences: A269250 A269251 A269252 * A269254 A269255 A269256


KEYWORD

sign


AUTHOR

Arkadiusz Wesolowski, Jul 09 2016


STATUS

approved



