A051834 -id:A051834 - cp's OEIS frontend

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

Jud McCranie, Dec 11 1999

nonn

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

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

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.

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

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

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

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

A051830 a(n) = Fibonacci(p(n)+1) mod p(n), where p(n) is the n-th prime.

Original entry on oeis.org

0, 0, 3, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1

Offset: 1

Jud McCranie, Dec 11 1999

nonn

Terms are 1 when p(n) == 1 or 4 (mod 5) and 0 when p(n) == 2 or 3 (mod 5).

			p(3) = 5, so a(3) = Fibonacci(5+1) mod 5 = 3.

Cf. A000040, A000045, A003631, A045468, A051834, A051831.

Table[Mod[Fibonacci[n+1],n],{n,Prime[Range[110]]}] (* Harvey P. Dale, Nov 27 2015 *)

a(n) = max(0, Legendre(5,prime(n))) for n >= 4, where Legendre is the Legendre symbol. - Haifeng Xu, Jan 31 2025

cp's OEIS Frontend

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

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