A116515 - OEIS

%I #4 Mar 31 2012 10:25:43

%S 1,2,4,2,1,4,4,1,2,1,1,4,2,2,2,4,1,4,2,1,4,1,2,4,4,1,2,2,4,4,2,1,4,1,

%T 4,1,4,2,2,4,1,1,1,4,4,1,1,2,2,1,4,1,2,1,4,2,4,1,4,2,2,4,2,1,4,4,1,4,

%U 2,1,4,1,2,4,1,2,4,4,2,2,1,4,1,4,1,2,2,4,1,2,2,1,2,1,1,2,1,1,2,1,2,4,2,2,1

%N a(n) = the period of the Fibonacci numbers modulo p divided by the smallest m such that p divides Fibonacci(m), where p is the n-th prime.

%C Conditions on p_n mod 4 and mod 5 restrict possible values of a(n). The unknown (?) case is p = 1 mod 4 and (5|p) = 1, equivalently, p = 1 or 9 mod 20, where {1, 2, 4} all occur.

%C Number of zeros in fundamental period of Fibonacci numbers mod prime(n). [From _T. D. Noe_, Jan 14 2009]

%F a(n) = A060305(n) / A001602(n). a(n) is always one of {1, 2, 4}.

%F a(n) = A001176(prime(n)) [From _T. D. Noe_, Jan 14 2009]

%e a(4) = 2, as 7 is the 4th prime, the Fibonacci numbers mod 7 have period 16, the first Fibonacci number divisible by 7 is F(8) = 21 = 3*7 and 16 / 8 = 2.

%e One period of the Fibonacci numbers mod 7 is 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, which has two zeros. Hence a(4)=2. [From _T. D. Noe_, Jan 14 2009]

%Y Cf. A060305, A001602.

%Y Cf. A112860, A053027, A053028 (primes producing 1, 2 and 4 zeros) [From _T. D. Noe_, Jan 14 2009]

%K easy,nonn

%O 1,2

%A _Nick Krempel_, Mar 24 2006