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%I #34 Feb 19 2021 14:15:38
%S 5,13,29,37,53,61,101,109,137,149,157,173,181,197,229,269,277,293,317,
%T 349,373,389,397,421,461,509,521,541,557,569,593,613,653,661,677,701,
%U 709,733,757,773,797,821,829,853,857,877,941,953,997,1013,1021,1061,1069
%N Primes such that z(p) is odd where z(n) is A214028(n).
%C From _Jianing Song_, Aug 13 2019: (Start)
%C Primes p with 4 zeros in a fundamental period of A000129 mod p, that is, primes p such that A214027(p) = 4.
%C For p > 2, p is in this sequence if and only if (all these conditions are equivalent):
%C (a) A175181(p) == 4 (mod 8);
%C (b) ord(p,1+sqrt(2)) == 4 (mod 8), where ord(p,u) is the smallest integer k > 0 such that (u^k - 1)/p is an algebraic integer;
%C (c) ord(p,3+2*sqrt(2)) == 2 (mod 4);
%C (d) A214028(p) is odd;
%C (e) ord(p,-(3+2*sqrt(2))) is odd.
%C In general, let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let pi(k) be the Pisano period of {x(n)} modulo k, i.e., pi(k) = min{l > 0 : x(n+l) == x(n) (mod k) for all n}, r(k) = min{l > 0 : k divides x(l)} and w(k) be the number of zeros in a fundamental period of {x(n)} modulo k. Let u = (m + sqrt(m^2+4))/2, p be an odd prime, then these conditions are equivalent:
%C (1) w(p) = 4;
%C (2) pi(p) == 4 (mod 8);
%C (3) ord(p,u) == 4 (mod 8);
%C (4) ord(p,u^2) == 2 (mod 4);
%C (5) r(p) is odd;
%C (6) ord(p,-u^2) is odd.
%C This can be shown by noting that pi(p) = p^c*ord(p,u) and r(p) = p^c*ord(p,-u^2) for some c (if p does not divide m^2 + 4 then c = 0, otherwise c = 1). Also, Pi(p) is always even, so ord(p,u) is always even.
%C This sequence contains all primes congruent to 5 modulo 8.
%C Conjecturely, this sequence has density 7/24 in the primes. (End)
%H Jianing Song, <a href="/A261580/b261580.txt">Table of n, a(n) for n = 1..1280</a>
%H Bernadette Faye, Florian Luca, <a href="http://arxiv.org/abs/1508.05714">Pell Numbers whose Euler Function is a Pell Number</a>, arXiv:1508.05714 [math.NT], 2015.
%e The smallest Pell number divisible by the prime 5 has index 3, which is odd, so 5 is in the sequence.
%t 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 *)
%o (PARI) pell(n) = polcoeff(Vec(x/(1-2*x-x^2) + O(x^(n+1))), n);
%o z(n) = {k=1; while (pell(k) % n, k++); k;}
%o lista(nn) = {forprime(p=2, nn, if (z(p) % 2, print1(p, ", ")););}
%o (PARI) forprime(p=2, 1100, if(A214027(p)==4, print1(p, ", "))) \\ _Jianing Song_, Aug 13 2019
%Y Cf. A000129, A214028, A261581.
%Y Cf. also A175181, A214027.
%Y Let {x(n)} be the sequence defined in the comment section.
%Y | m=1 | m=2 | m=3
%Y Primes p such that w(p) = 1 | A112860* | A309580 | A309586
%Y Primes p such that w(p) = 2 | A053027 | A309581 | A309587
%Y Primes p such that w(p) = 4 | A053028 | this seq | A309588
%Y Numbers k such that w(k) = 1 | A053031 | A309583 | A309591
%Y Numbers k such that w(k) = 2 | A053030 | A309584 | A309592
%Y Numbers k such that w(k) = 4 | A053029 | A309585 | A309593
%Y * and also A053032 U {2}
%K nonn
%O 1,1
%A _Michel Marcus_, Aug 25 2015
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