A309581 Primes p with 2 zeros in a fundamental period of A000129 mod p.

3, 11, 17, 19, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 331, 337, 347, 379, 401, 419, 433, 443, 449, 467, 491, 499, 523, 547, 563, 571, 577, 587, 601, 617, 619, 641, 643, 659, 673, 683, 691

Offset

1,1

Comments

Primes p such that A214027(p) = 2.

For p > 2, p is in this sequence if and only if 8 divides A175181(p), and if and only if 4 divides A214028(p). For a proof of the equivalence between A214027(p) = 2 and 4 dividing A214028(p), see Section 2 of my link below.

This sequence contains all primes congruent to 3 modulo 8. This corresponds to case (2) for k = 6 in the Conclusion of Section 1 of my link below.

Conjecturely, this sequence has density 5/12 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..1600

Jianing Song, Lucas sequences and entry point modulo p

Prog

(PARI) forprime(p=2, 700, if(A214027(p)==2, print1(p, ", ")))

Crossrefs

Cf. A175181, A214028.

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 | this seq | 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: A210311 A033200 A369171 * A291277 A191375 A260793

Adjacent sequences: A309578 A309579 A309580 * A309582 A309583 A309584

Keyword

nonn

Author

Jianing Song, Aug 10 2019

Status

approved