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%I #41 Feb 02 2022 23:51:16
%S 11,19,29,31,59,71,79,101,131,139,151,179,181,191,199,211,229,239,251,
%T 271,311,331,349,359,379,419,431,439,461,479,491,499,509,521,541,571,
%U 599,619,631,659,691,709,719,739,751,809,811,839,859,911,919,941,971
%N Odd primes p with one zero in Fibonacci numbers mod p.
%C Also, odd primes that divide Lucas numbers of odd index. - _T. D. Noe_, Jul 25 2003
%C From _Charles R Greathouse IV_, Dec 14 2016: (Start)
%C It seems that this sequence contains about 1/3 of the primes. In particular, members of this sequence constitute:
%C 35 of the first 10^2 primes
%C 330 of the first 10^3 primes
%C 3328 of the first 10^4 primes
%C 33371 of the first 10^5 primes
%C 333329 of the first 10^6 primes
%C 3333720 of the first 10^7 primes
%C 33333463 of the first 10^8 primes
%C etc. (End)
%C 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 Odd primes in A053031. - _Jianing Song_, Jun 19 2019
%H T. D. Noe, <a href="/A053032/b053032.txt">Table of n, a(n) for n = 1..1000</a>
%H C. Ballot and M. Elia, <a href="http://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 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 M. Renault, <a href="http://webspace.ship.edu/msrenault/fibonacci/fib.htm">Fibonacci sequence modulo m</a>
%F 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 From _Michael B. Porter_, Jan 25 2019: (Start)
%e 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 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 (End)
%t 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 (PARI) fibmod(n,m)=(Mod([1, 1; 1, 0], m)^n)[1, 2]
%o 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 Cf. A001176, A053031. See A112860 for another version.
%Y Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)), A053028 (primes dividing no Lucas number), A053027 (primes dividing Lucas numbers of even index).
%K nonn
%O 1,1
%A _Henry Bottomley_, Feb 23 2000
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