cp's OEIS Frontend

A053032 Odd primes p with one zero in Fibonacci numbers mod p.

%I A053032 #53 Jan 05 2025 19:51:36
%S A053032 11,19,29,31,59,71,79,101,131,139,151,179,181,191,199,211,229,239,251,
%T A053032 271,311,331,349,359,379,419,431,439,461,479,491,499,509,521,541,571,
%U A053032 599,619,631,659,691,709,719,739,751,809,811,839,859,911,919,941,971
%N A053032 Odd primes p with one zero in Fibonacci numbers mod p.
%C A053032 Also, odd primes that divide Lucas numbers of odd index. - _T. D. Noe_, Jul 25 2003
%C A053032 From _Charles R Greathouse IV_, Dec 14 2016: (Start)
%C A053032 It seems that this sequence contains about 1/3 of the primes. In particular, members of this sequence constitute:
%C A053032         35 of the first 10^2 primes
%C A053032        330 of the first 10^3 primes
%C A053032       3328 of the first 10^4 primes
%C A053032      33371 of the first 10^5 primes
%C A053032     333329 of the first 10^6 primes
%C A053032    3333720 of the first 10^7 primes
%C A053032   33333463 of the first 10^8 primes
%C A053032 etc. (End)
%C A053032 Of the Fibonacci-like sequences modulo a prime p that are not A000004, one of them has a period length less than A001175(p) if and only if p = 5 or p is in this sequence. - _Isaac Saffold_, Dec 18 2018
%C A053032 Odd primes in A053031. - _Jianing Song_, Jun 19 2019
%H A053032 T. D. Noe, <a href="/A053032/b053032.txt">Table of n, a(n) for n = 1..1000</a>
%H A053032 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 B1).
%H A053032 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 A053032 M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">Fibonacci sequence modulo m</a>
%F A053032 A prime p = prime(i) is in this sequence if p > 2 and A001602(i)/2 is odd. - _T. D. Noe_, Jul 25 2003
%e A053032 From _Michael B. Porter_, Jan 25 2019: (Start)
%e A053032 The Fibonacci numbers (mod 7) repeat the pattern 0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1. Since there are two zeros, 7 is not in the sequence.
%e A053032 The Fibonacci numbers (mod 11) repeat the pattern 0, 1, 1, 2, 3, 5, 8, 2, 10, 1 which has only one zero, so 11 is in the sequence.
%e A053032 (End)
%t A053032 Prime@ Rest@ Position[Table[Count[Drop[NestWhile[Append[#, Mod[Total@ Take[#, -2], n]] &, {1, 1}, If[Length@ # < 3, True, Take[#, -2] != {1, 1}] &], -2], 0], {n, Prime@ Range@ 168}], 1][[All, 1]] (* _Michael De Vlieger_, Aug 08 2018 *)
%o A053032 (PARI) fibmod(n,m)=(Mod([1, 1; 1, 0], m)^n)[1, 2]
%o A053032 is(n)=my(k=n+[0, -1, 1, 1, -1][n%5+1]); k>>=valuation(k,2)-1; fibmod(k,n)==0 && fibmod(k/2,n) && isprime(n) \\ _Charles R Greathouse IV_, Dec 14 2016
%Y A053032 Cf. A001175, A001177. See A112860 for another version.
%Y A053032 Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)).
%Y A053032 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 A053032                              |    m=1     |   m=2   |   m=3
%Y A053032 -----------------------------+------------+---------+---------
%Y A053032 The sequence {x(n)}          | A000045    | A000129 | A006190
%Y A053032 The sequence {w(k)}          | A001176    | A214027 | A322906
%Y A053032 Primes p such that w(p) = 1  | A112860*   | A309580 | A309586
%Y A053032 Primes p such that w(p) = 2  | A053027**  | A309581 | A309587
%Y A053032 Primes p such that w(p) = 4  | A053028*** | A261580 | A309588
%Y A053032 Numbers k such that w(k) = 1 | A053031    | A309583 | A309591
%Y A053032 Numbers k such that w(k) = 2 | A053030    | A309584 | A309592
%Y A053032 Numbers k such that w(k) = 4 | A053029    | A309585 | A309593
%Y A053032 * and also this sequence U {2}
%Y A053032 ** also primes dividing Lucas numbers of even index
%Y A053032 *** also primes dividing no Lucas number
%K A053032 nonn
%O A053032 1,1
%A A053032 _Henry Bottomley_, Feb 23 2000