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A053032 Odd primes p with one zero in Fibonacci numbers mod p. 11
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)

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

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. See A112860 for another version.

Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)), A053029 (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 December 17 05:36 EST 2018. Contains 318192 sequences. (Running on oeis4.)