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A053028
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Odd primes p with 4 zeros in Fibonacci numbers mod p.
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14
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5, 13, 17, 37, 53, 61, 73, 89, 97, 109, 113, 137, 149, 157, 173, 193, 197, 233, 257, 269, 277, 293, 313, 317, 337, 353, 373, 389, 397, 421, 433, 457, 557, 577, 593, 613, 617, 653, 661, 673, 677, 701, 733, 757, 761, 773, 797, 821, 829, 853, 857, 877, 937, 953
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OFFSET
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1,1
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COMMENTS
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Also, primes that do not divide any Lucas number. - T. D. Noe, Jul 25 2003
Although every prime divides some Fibonacci number, this is not true for the Lucas numbers. In fact, exactly 1/3 of all primes do not divide any Lucas number. See Lagarias and Moree for more details. The Lucas numbers separate the primes into three disjoint sets: (A053028) primes that do not divide any Lucas number, (A053027) primes that divide Lucas numbers of even index and (A053032) primes that divide Lucas numbers of odd index. - T. D. Noe, Jul 25 2003; revised N. J. A. Sloane, Feb 21, 2004
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REFERENCES
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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 B2).
L. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118 (1985), 449-461.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
M. Renault, Fibonacci sequence modulo m
Eric Weisstein's World of Mathematics, Lucas Number
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FORMULA
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A prime p = prime(i) is in this sequence if p > 2 and A001602(i) is odd. - T. D. Noe, Jul 25 2003
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MATHEMATICA
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Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; badP={}; Do[p=Prime[n]; k=1; While[k<p&&Mod[Lucas[k], p]>0, k++ ]; If[k==p, AppendTo[badP, p]], {n, 200}]; badP
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CROSSREFS
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Cf. A001176.
Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)), A053027, A053032.
Sequence in context: A211425 A191108 A216575 * A189411 A188131 A172459
Adjacent sequences: A053025 A053026 A053027 * A053029 A053030 A053031
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley, Feb 23 2000
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STATUS
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approved
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