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, since (k+2)/2 = 4 is a square, this sequence has density 5/12 in the primes; see the end of Section 1 of my link. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

The conjecture above is an analog of Hasse's result that the set {p prime : ord(2,p) is odd} has density 7/24 in the primes, where ord(a,m) is the multiplicative order of a modulo m; see A014663. - Jianing Song, Jun 26 2025

Links

Jianing Song, Table of n, a(n) for n = 1..1600

Jianing Song, Entry points and periods of Lucas sequences

Prog

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

Crossrefs

Cf. A175181, A214028.

For a list of sequences related to the numbers of zeros in a fundamental period of {x(n)}, where {x(n)} is a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n), see A053032.

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