OFFSET
1,1
COMMENTS
Also, odd primes that divide Lucas numbers of even index. - T. D. Noe, Jul 25 2003
Primes in A053030. - Jianing Song, Jun 19 2019
From Jianing Song, Jun 16 2024: (Start)
Primes p such that A001176(p) = 2.
For p > 2, p is in this sequence if and only if 8 divides of A001175(p), and if and only if 4 divides A001177(p). For a proof of the equivalence between A001176(p) = 2 and 4 dividing A001177(p), see Section 2 of my link below.
This sequence contains all primes congruent to 3, 7 (mod 20). This corresponds to case (2) for k = 3 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]
LINKS
T. D. Noe, Table of n, a(n) for n=1..1000
C. Ballot and M. Elia, Rank and period of primes in the Fibonacci sequence; a trichotomy, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B3).
Nicholas Bragman and Eric Rowland, Limiting density of the Fibonacci sequence modulo powers of p, arXiv:2202.00704 [math.NT], 2022.
M. Renault, Fibonacci sequence modulo m
Jianing Song, Lucas sequences and entry point modulo p
FORMULA
CROSSREFS
Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)).
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
-----------------------------+-----------+---------+---------
* and also A053032 (primes dividing Lucas numbers of odd index) U {2}
** also primes dividing no Lucas number
KEYWORD
nonn
AUTHOR
Henry Bottomley, Feb 23 2000
STATUS
approved