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A053028 Odd primes p with 4 zeros in Fibonacci numbers mod p. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

REFERENCES

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.

LINKS

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

FORMULA

A prime p = prime(i) is in this sequence if p > 2 and A001602(i) is odd. - T. D. Noe, Jul 25 2003

MATHEMATICA

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

CROSSREFS

Cf. A001176.

Cf. A000204 (Lucas numbers), A001602 (index of the smallest Fibonacci number divisible by prime(n)), A053027, A053032.

Sequence in context: A245906 A191108 A216575 * A189411 A248980 A188131

Adjacent sequences:  A053025 A053026 A053027 * A053029 A053030 A053031

KEYWORD

nonn

AUTHOR

Henry Bottomley, Feb 23 2000

STATUS

approved

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Last modified December 17 18:04 EST 2014. Contains 252034 sequences.