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 A053032 Odd primes p with one zero in Fibonacci numbers mod p. 22
 11, 19, 29, 31, 59, 71, 79, 101, 131, 139, 151, 179, 181, 191, 199, 211, 229, 239, 251, 271, 311, 331, 349, 359, 379, 419, 431, 439, 461, 479, 491, 499, 509, 521, 541, 571, 599, 619, 631, 659, 691, 709, 719, 739, 751, 809, 811, 839, 859, 911, 919, 941, 971 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Also, odd primes that divide Lucas numbers of odd index. - T. D. Noe, Jul 25 2003 From Charles R Greathouse IV, Dec 14 2016: (Start) It seems that this sequence contains about 1/3 of the primes. In particular, members of this sequence constitute:         35 of the first 10^2 primes        330 of the first 10^3 primes       3328 of the first 10^4 primes      33371 of the first 10^5 primes     333329 of the first 10^6 primes    3333720 of the first 10^7 primes   33333463 of the first 10^8 primes etc. (End) 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 Odd primes in A053031. - Jianing Song, Jun 19 2019 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 B1). M. Renault, Fibonacci sequence modulo m FORMULA A prime p = prime(i) is in this sequence if p > 2 and A001602(i)/2 is odd. - T. D. Noe, Jul 25 2003 EXAMPLE From Michael B. Porter, Jan 25 2019: (Start) 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. 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. (End) MATHEMATICA 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 *) PROG (PARI) fibmod(n, m)=(Mod([1, 1; 1, 0], m)^n)[1, 2] 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 CROSSREFS Cf. A001176, A053031. See A112860 for another version. 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). Sequence in context: A045468 A196095 A268271 * A277123 A034099 A034109 Adjacent sequences:  A053029 A053030 A053031 * A053033 A053034 A053035 KEYWORD nonn AUTHOR Henry Bottomley, Feb 23 2000 STATUS approved

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Last modified October 23 17:32 EDT 2019. Contains 328373 sequences. (Running on oeis4.)