A261580 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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

Cf. A214028, A261581.

Cf. also A175181.

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.

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

-----------------------------+----------+----------+---------

The sequence {x(n)} | A000045 | A000129 | A006190

The sequence {w(k)} | A001176 | A214027 | A322906

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