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A053032
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Odd primes p with one zero in Fibonacci numbers mod p.
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24
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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
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OFFSET
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1,1
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COMMENTS
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Also, odd primes that divide Lucas numbers of odd index. - T. D. Noe, Jul 25 2003
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
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LINKS
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FORMULA
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A prime p = prime(i) is in this sequence if p > 2 and A001602(i)/2 is odd. - T. D. Noe, Jul 25 2003
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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