A309586 - OEIS

A309586

Primes p with 1 zero in a fundamental period of A006190 mod p.

2, 3, 23, 43, 53, 61, 79, 101, 103, 107, 127, 131, 139, 173, 179, 191, 199, 211, 251, 263, 277, 283, 311, 347, 367, 419, 433, 439, 443, 467, 491, 503, 523, 547, 563, 569, 571, 599, 607, 647, 659, 677, 719, 727, 751, 757, 823, 829, 859, 881, 883, 887, 907

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

OFFSET

1,1

COMMENTS

Primes p such that A322906(p) = 1.

For p > 2, p is in this sequence if and only if A175182(p) == 2 (mod 4), and if and only if A322907(p) == 2 (mod 4). For a proof of the equivalence between A322906(p) = 1 and A322907(p) == 2 (mod 4), see Section 2 of my link below.

This sequence contains all primes congruent to 3, 23, 27, 35, 43, 51 modulo 52. This corresponds to case (3) for k = 11 in the Conclusion of Section 1 of my link below.

Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

LINKS

Jianing Song, Table of n, a(n) for n = 1..1200

Jianing Song, Lucas sequences and entry point modulo p

PROG

(PARI) forprime(p=2, 900, if(A322906(p)==1, print1(p, ", ")))

CROSSREFS

Cf. A175182, A322907.

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 | this seq

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

Primes p such that w(p) = 4 | A053028 | A261580 | 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: A215305 A215282 A356498 * A090708 A359406 A199342

Adjacent sequences: A309583 A309584 A309585 * A309587 A309588 A309589

KEYWORD

nonn

AUTHOR

Jianing Song, Aug 10 2019

STATUS

approved