

A246556


a(n) = smallest prime which divides Pell(n) = A000129(n) but does not divide any Pell(k) for k<n, or 1 if no such prime exists.


4



2, 5, 3, 29, 7, 13, 17, 197, 41, 5741, 11, 33461, 239, 269, 577, 137, 199, 37, 19, 45697, 23, 229, 1153, 1549, 79, 53, 113, 44560482149, 31, 61, 665857, 52734529, 103, 1800193921, 73, 593, 9369319, 389, 241, 1746860020068409, 4663, 11437, 43, 6481, 47, 3761, 97, 293, 45245801, 101, 22307, 68480406462161287469, 7761799, 109, 1535466241
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

2,1


COMMENTS

First differs from A264137 (Largest prime factor of the nth Pell number) at n=17; see Example section.  Jon E. Schoenfield, Dec 10 2016


LINKS

Table of n, a(n) for n = 2..630
R. D. Carmichael, On the numerical factors of the arithmetic forms α^n ± β^n, Annals of Math., 15 (1/4) (1913), 3070.


FORMULA

a(n) >= 2 for all n >= 2, by Carmichael's theorem.  Jonathan Sondow, Dec 08 2017


EXAMPLE

a(2) = 2 because Pell(2) = 2 and Pell(k) < 2 for k < 2.
a(4) = 3 because Pell(4) = 12 = 2^2 * 3, but 2 is not a primitive prime factor since Pell(2) = 2, so therefore 3 is the primitive prime factor.
a(5) = 29 because Pell(5) = 29, which is prime.
a(6) = 7 because Pell(6) = 70 = 2 * 5 * 7, but neither 2 nor 5 is a primitive prime factor, so therefore 7 is the primitive prime factor.
a(17) = 137 because Pell(17) = 1136689 = 137 * 8297, and both of them are primitive factors, we choose the smallest. (Pell(17) is the smallest Pell number with more than one primitive prime factor.)


MATHEMATICA

prms={}; Table[f=First/@FactorInteger[Pell[n]]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, First[p]], {n, 36}]


CROSSREFS

Cf. A001578 (for Fibonacci(n)), A000129 (Pell numbers), A008555, A086383, A096650, A120947, A175181, A214028, A264137.
Sequence in context: A097754 A122992 A051497 * A264137 A308949 A109734
Adjacent sequences: A246553 A246554 A246555 * A246557 A246558 A246559


KEYWORD

nonn


AUTHOR

Eric Chen, Nov 15 2014


EXTENSIONS

Edited by N. J. A. Sloane, Nov 29 2014
Terms up to a(612) in bfile added by Sean A. Irvine, Sep 23 2019
Terms a(613)a(630) in bfile added by Max Alekseyev, Aug 26 2021


STATUS

approved



