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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 n-th 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), 30-70.

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 b-file added by Sean A. Irvine, Sep 23 2019

Terms a(613)-a(630) in b-file added by Max Alekseyev, Aug 26 2021

STATUS

approved

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Last modified September 28 14:44 EDT 2022. Contains 357073 sequences. (Running on oeis4.)