A053027 Odd primes p with 2 zeros in Fibonacci numbers mod p.

3, 7, 23, 41, 43, 47, 67, 83, 103, 107, 127, 163, 167, 223, 227, 241, 263, 281, 283, 307, 347, 367, 383, 401, 409, 443, 449, 463, 467, 487, 503, 523, 547, 563, 569, 587, 601, 607, 641, 643, 647, 683, 727, 743, 769, 787, 823, 827, 863, 881, 883, 887, 907, 929

Offset: 1

Author

Henry Bottomley, Feb 23 2000

Keywords

• nonn

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

Crossrefs

Cf. A001175, A001177.

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

-----------------------------+-----------+---------+---------

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 | this seq | 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 (primes dividing Lucas numbers of odd index) U {2}

** also primes dividing no Lucas number

Formula

A prime p = prime(i) is in this sequence if p > 2 and A001602(i)/2 is even. - T. D. Noe, Jul 25 2003