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A001602
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Fibonacci entry points: a(n) = smallest m > 0 such that the n-th prime divides Fibonacci(m).
(Formerly M2310 N0912)
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51
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3, 4, 5, 8, 10, 7, 9, 18, 24, 14, 30, 19, 20, 44, 16, 27, 58, 15, 68, 70, 37, 78, 84, 11, 49, 50, 104, 36, 27, 19, 128, 130, 69, 46, 37, 50, 79, 164, 168, 87, 178, 90, 190, 97, 99, 22, 42, 224, 228, 114, 13, 238, 120, 250, 129, 88, 67, 270, 139, 28, 284, 147, 44, 310
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OFFSET
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1,1
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COMMENTS
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"[a(n)] is called by Lucas the rank of apparition of p and we know it is a divisor of, or equal to prime(n)-1 or prime(n)+1" - Vajda, p. 84. (Note that a(3)=5. This is the only exception.) - Chris K. Caldwell, Nov 03 2008
Every number except 1, 2, 6 and 12 eventually occurs in this sequence. See also A086597(n), the number of primitive prime factors of Fibonacci(n). - T. D. Noe, Jun 13 2008
For each prime p we have an infinite sequence of integers, F(i*a(n))/p, i=1,2,... See also A236479. For primes p >= 3 and exponents j >= 2, with k = a(n) and p = p(n), it appears that F(k*i*p^(j-1))/p^j is an integer, for i >= 0. For p = 2, F(k*i*p^(j-1))/p^(j+1) = integer. - Richard R. Forberg, Jan 26-29 2014 [Comments revised by N. J. A. Sloane, Sep 24 2015]
Let p=prime(n). a(n) is also a divisor of (p-1)/2 (if p mod 5 == 1 or 4) or (p+1)/2 (if p mod 5 == 2 or 3) if and only if p mod 4 = 1. - Azuma Seiichi, Jul 29 2014
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.
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LINKS
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Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy, pp. 2-3 missing] See p. 7.
D. Lind et al., Tables of Fibonacci entry points, part 2, reviewed in, Math. Comp., 20 (1966), 618-619.
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FORMULA
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EXAMPLE
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The 5th prime is 11 and 11 first divides Fib(10)=55, so a(5) = 10.
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MAPLE
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local i, p;
p := ithprime(n);
for i from 1 do
if modp(combinat[fibonacci](i), p) = 0 then
return i;
end if;
end do:
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MATHEMATICA
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Table[k=1; While[!Divisible[Fibonacci[k], Prime[n]], k++]; k, {n, 70}] (* Harvey P. Dale, Feb 15 2012 *)
(* a fast, but more complicated method *) MatrixPowerMod[mat_, n_, m_Integer] := Mod[Fold[Mod[If[#2 == 1, #1.#1.mat, #1.#1], m] &, mat, Rest[IntegerDigits[n, 2]]], m]; FibMatrix[n_Integer, m_Integer] := MatrixPowerMod[{{0, 1}, {1, 1}}, n, m]; FibEntryPointPrime[p_Integer] := Module[{n, d, k}, If[PrimeQ[p], n = p - JacobiSymbol[p, 5]; d = Divisors[n]; k = 1; While[FibMatrix[d[[k]], p][[1, 2]] > 0, k++]; d[[k]]]]; SetAttributes[FibEntryPointPrime, Listable]; FibEntryPointPrime[Prime[Range[10000]]] (* T. D. Noe, Jan 03 2013 *)
With[{nn=70, t=Table[{n, Fibonacci[n]}, {n, 500}]}, Transpose[ Flatten[ Table[ Select[t, Divisible[#[[2]], Prime[i]]&, 1], {i, nn}], 1]][[1]]] (* Harvey P. Dale, May 31 2014 *)
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PROG
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(Haskell)
import Data.List (findIndex)
import Data.Maybe (fromJust)
a001602 n = (+ 1) $ fromJust $
findIndex ((== 0) . (`mod` a000040 n)) $ tail a000045_list
(PARI) a(n)=if(n==3, 5, my(p=prime(n)); fordiv(p^2-1, d, if(fibonacci(d)%p==0, return(d)))) \\ Charles R Greathouse IV, Jul 17 2012
(PARI) do(p)=my(k=p+[0, -1, 1, 1, -1][p%5+1], f=factor(k)); for(i=1, #f[, 1], for(j=1, f[i, 2], if((Mod([1, 1; 1, 0], p)^(k/f[i, 1]))[1, 2], break); k/=f[i, 1])); k
a(n)=do(prime(n))
(Python)
from sympy.ntheory.generate import prime
a, b, i, p = 0, 1, 1, prime(n)
while b % p:
a, b, i = b, (a+b) % p, i+1
return i # Chai Wah Wu, Nov 03 2015, revised Apr 04 2016.
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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