A051831 a(n) = Fibonacci(prime(n)) mod prime(n), where prime(n) is the n-th prime.

1, 2, 0, 6, 1, 12, 16, 1, 22, 1, 1, 36, 1, 42, 46, 52, 1, 1, 66, 1, 72, 1, 82, 1, 96, 1, 102, 106, 1, 112, 126, 1, 136, 1, 1, 1, 156, 162, 166, 172, 1, 1, 1, 192, 196, 1, 1, 222, 226, 1, 232, 1, 1, 1, 256, 262, 1, 1, 276, 1, 282, 292, 306, 1, 312, 316, 1, 336, 346, 1, 352, 1

Offset: 1

Author

Jud McCranie, Dec 11 1999

Keywords

• nonn

Comments

Terms are 1 when prime(n) == 1 or 4 mod 5, terms are prime(n)-1 when prime(n) == 2 or 3 mod 5.

In general, it appears that Fibonacci(k*p) mod p = Fibonacci(k) or p-Fibonacci(k) for prime p > Fibonacci(k). For example Fibonacci(8*29) mod 29 = 21. - Gary Detlefs, May 28 2014

Examples

			prime(3) = 5, fibonacci(5) = 5 == 0 mod 5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• R. Peele and P. Stanica, Matrix powers of column-justified Pascal triangles and Fibonacci sequences, arXiv:math/0010186 [math.CO], 2000.

Crossrefs

Cf. A000045, A003631, A045468, A051834, A051830, A236395.

Programs

Maple

p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, ithprime(n)$2)[1, 2]:
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 10 2015

Mathematica

Mod[Fibonacci[Prime[#]],Prime[#]]&/@Range[75] (* Harvey P. Dale, Jan 14 2011 *)

PARI

vector(80, n, fibonacci(prime(n)) % prime(n)) \\ Michel Marcus, Jul 15 2015