OFFSET
1,1
COMMENTS
From Jianing Song, Aug 13 2019: (Start)
Primes p with 4 zeros in a fundamental period of A000129 mod p, that is, primes p such that A214027(p) = 4. For a proof of the equivalence between A214027(p) = 4 and A214028(p) being odd, see Section 2 of my link below.
For p > 2, p is in this sequence if and only if A175181(p) == 4 (mod 8).
This sequence contains all primes congruent to 5 modulo 8. This corresponds to case (1) for k = 6 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 7/24 in the primes. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 20 2024]
LINKS
Jianing Song, Table of n, a(n) for n = 1..1280
Bernadette Faye and Florian Luca, Pell Numbers whose Euler Function is a Pell Number, arXiv:1508.05714 [math.NT], 2015.
Jianing Song, Lucas sequences and entry point modulo p
EXAMPLE
The smallest Pell number divisible by the prime 5 has index 3, which is odd, so 5 is in the sequence.
MATHEMATICA
f[n_] := Block[{k = 1}, While[Mod[Simplify[((1 + Sqrt@ 2)^k - (1 - Sqrt@ 2)^k)/(2 Sqrt@ 2)], n] != 0, k++]; k]; Select[Prime@ Range@ 180, OddQ@ f@ # &] (* Michael De Vlieger, Aug 25 2015 *)
PROG
(PARI) pell(n) = polcoeff(Vec(x/(1-2*x-x^2) + O(x^(n+1))), n);
z(n) = {k=1; while (pell(k) % n, k++); k; }
lista(nn) = {forprime(p=2, nn, if (z(p) % 2, print1(p, ", ")); ); }
(PARI) forprime(p=2, 1100, if(A214027(p)==4, print1(p, ", "))) \\ Jianing Song, Aug 13 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Aug 25 2015
STATUS
approved