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

%I A053027 #41 Jan 05 2025 19:51:36
%S A053027 3,7,23,41,43,47,67,83,103,107,127,163,167,223,227,241,263,281,283,
%T A053027 307,347,367,383,401,409,443,449,463,467,487,503,523,547,563,569,587,
%U A053027 601,607,641,643,647,683,727,743,769,787,823,827,863,881,883,887,907,929
%N A053027 Odd primes p with 2 zeros in Fibonacci numbers mod p.
%C A053027 Also, odd primes that divide Lucas numbers of even index. - _T. D. Noe_, Jul 25 2003
%C A053027 Primes in A053030. - _Jianing Song_, Jun 19 2019
%C A053027 From _Jianing Song_, Jun 16 2024: (Start)
%C A053027 Primes p such that A001176(p) = 2.
%C A053027 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.
%C A053027 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.
%C A053027 Conjecturely, this sequence has density 1/3 in the primes. (End) [Comment rewritten by _Jianing Song_, Jun 16 2024 and Jun 25 2024]
%H A053027 T. D. Noe, <a href="/A053027/b053027.txt">Table of n, a(n) for n=1..1000</a>
%H A053027 C. Ballot and M. Elia, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/45-1/quartballot01_2007.pdf">Rank and period of primes in the Fibonacci sequence; a trichotomy</a>, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B3).
%H A053027 Nicholas Bragman and Eric Rowland, <a href="https://arxiv.org/abs/2202.00704">Limiting density of the Fibonacci sequence modulo powers of p</a>, arXiv:2202.00704 [math.NT], 2022.
%H A053027 M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">Fibonacci sequence modulo m</a>
%H A053027 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%F A053027 A prime p = prime(i) is in this sequence if p > 2 and A001602(i)/2 is even. - _T. D. Noe_, Jul 25 2003
%Y A053027 Cf. A001175, A001177.
%Y A053027 Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)).
%Y A053027 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.
%Y A053027                              |    m=1    |   m=2   |   m=3
%Y A053027 -----------------------------+-----------+---------+---------
%Y A053027 The sequence {x(n)}          | A000045   | A000129 | A006190
%Y A053027 The sequence {w(k)}          | A001176   | A214027 | A322906
%Y A053027 Primes p such that w(p) = 1  | A112860*  | A309580 | A309586
%Y A053027 Primes p such that w(p) = 2  | this seq  | A309581 | A309587
%Y A053027 Primes p such that w(p) = 4  | A053028** | A261580 | A309588
%Y A053027 Numbers k such that w(k) = 1 | A053031   | A309583 | A309591
%Y A053027 Numbers k such that w(k) = 2 | A053030   | A309584 | A309592
%Y A053027 Numbers k such that w(k) = 4 | A053029   | A309585 | A309593
%Y A053027 * and also A053032 (primes dividing Lucas numbers of odd index) U {2}
%Y A053027 ** also primes dividing no Lucas number
%K A053027 nonn
%O A053027 1,1
%A A053027 _Henry Bottomley_, Feb 23 2000