A120947 -id:A120947 - cp's OEIS frontend

A246556 a(n) = smallest prime which divides Pell(n) = A000129(n) but does not divide any Pell(k) for k

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

Offset: 2

Eric Chen, Nov 15 2014

nonn

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

			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.)

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.

Cf. A001578 (for Fibonacci(n)), A000129 (Pell numbers), A008555, A086383, A096650, A120947, A175181, A214028, A264137.

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

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

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

A239111 Smallest Pell number (see A000129) divisible by n-th prime.

Original entry on oeis.org

2, 12, 5, 70, 13860, 169, 408, 15994428, 93222358, 29, 107578520350, 6625109, 2378, 24580185800219268, 143263821649299118, 7645370045, 15994428, 259717522849, 37774750930342781945186508, 220167382952941249990598278, 21300003689580, 3166815962, 50305164660422142002238655969020

Offset: 1

N. J. A. Sloane, Mar 27 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..350
J. L. Schiffman, Exploring the Fibonacci sequence of order two with CAS technology, Paper C027, Electronic Proceedings of the Twenty-fourth Annual International Conference on Technology in Collegiate Mathematics, Orlando, Florida, March 22-25, 2012.

Cf. A120947.

p:= proc(n) p(n):=`if`(n<2, n, 2*p(n-1)+p(n-2)) end:
a:= proc(n) local k, t; t:= ithprime(n);
      for k while irem(p(k), t)>0 do od; p(k)
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, Mar 28 2014

p[n_] := p[n] = If[n<2, n, 2*p[n-1] + p[n-2]]; a[n_] := With[{t = Prime[n]} , For[k=1, Mod[p[k], t] > 0, k++]; p[k]]; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Feb 14 2016, after Alois P. Heinz *)

cp's OEIS Frontend

A246556 a(n) = smallest prime which divides Pell(n) = A000129(n) but does not divide any Pell(k) for k

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