A261580 Primes such that z(p) is odd where z(n) is A214028(n).

5, 13, 29, 37, 53, 61, 101, 109, 137, 149, 157, 173, 181, 197, 229, 269, 277, 293, 317, 349, 373, 389, 397, 421, 461, 509, 521, 541, 557, 569, 593, 613, 653, 661, 677, 701, 709, 733, 757, 773, 797, 821, 829, 853, 857, 877, 941, 953, 997, 1013, 1021, 1061, 1069

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 p > 2, p is in this sequence if and only if (all these conditions are equivalent):

(a) A175181(p) == 4 (mod 8);

(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) ord(p,3+2*sqrt(2)) == 2 (mod 4);

(d) A214028(p) is odd;

(e) ord(p,-(3+2*sqrt(2))) is odd.

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:

(1) w(p) = 4;

(2) pi(p) == 4 (mod 8);

(3) ord(p,u) == 4 (mod 8);

(4) ord(p,u^2) == 2 (mod 4);

(5) r(p) is odd;

(6) ord(p,-u^2) is odd.

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.

This sequence contains all primes congruent to 5 modulo 8.

Conjecturely, this sequence has density 7/24 in the primes. (End)

Links

Jianing Song, Table of n, a(n) for n = 1..1280

Bernadette Faye, Florian Luca, Pell Numbers whose Euler Function is a Pell Number, arXiv:1508.05714 [math.NT], 2015.

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

Cf. A000129, A214028, A261581.

Cf. also A175181, A214027.

Let {x(n)} be the sequence defined in the comment section.

| m=1 | m=2 | m=3

Primes p such that w(p) = 1 | A112860* | A309580 | A309586

Primes p such that w(p) = 2 | A053027 | A309581 | A309587

Primes p such that w(p) = 4 | A053028 | this seq | A309588

Numbers k such that w(k) = 1 | A053031 | A309583 | A309591

Numbers k such that w(k) = 2 | A053030 | A309584 | A309592

Numbers k such that w(k) = 4 | A053029 | A309585 | A309593

* and also A053032 U {2}

Sequence in context: A152658 A347836 A100877 * A007521 A294919 A213050

Adjacent sequences: A261577 A261578 A261579 * A261581 A261582 A261583

Keyword

nonn

Author

Michel Marcus, Aug 25 2015

Status

approved