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

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]

References

Ballot, Christian. "Prime Factors of Fibonacci-Related Recurrences." The Fibonacci Quarterly 63.2 (2025): 178-206.

Links

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

Jianing Song, Entry points and periods of Lucas sequences

Prog

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

Crossrefs

Cf. A175182, A322907.

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: A215305 A215282 A356498 * A386142 A090708 A359406

Adjacent sequences: A309583 A309584 A309585 * A309587 A309588 A309589

Keyword

nonn

Author

Jianing Song, Aug 10 2019

Status

approved