A222361 Fibonacci-Legendre quotients: (Fibonacci(p) - L(p/5)) / p, where p = prime(n) and L(p/5) is the Legendre symbol.

1, 1, 1, 2, 8, 18, 94, 220, 1246, 17732, 43428, 652914, 4038540, 10081266, 63217342, 1005967758, 16215627560, 41061160360, 670829406162, 4338894664368, 11048157986978, 183194101578180, 1195118711985006, 19999768719154092, 862073644225241474, 5674731128849674100, 14568160545698020226, 96118885585174929102, 247025215671874138312, 1633201998168434481118

Offset: 1

Author

Jonathan Sondow, Feb 23 2013

Keywords

• nonn

Comments

Fibonacci(p) == L(p/5) mod p, where the Legendre symbol L(p/5) equals 0, +1, -1 according as p = 5, 5*k+-1, 5*k+-2 for some k.

Not to be confused with Fibonacci(p - L(p,5)) / p, which is A092330.

Examples

			Prime(4) = 7, so a(4) = (Fibonacci(7)-L(7/5))/7 = (13-(-1))/7 = 14/7 = 2.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Wikipedia, Prime divisors of Fibonacci numbers

Crossrefs

Cf. A000045, A092330.

Programs

Mathematica

Table[p = Prime[n]; (Fibonacci[p] - JacobiSymbol[p, 5])/p, {n, 1, 30}]

Formula

For n>=4, a(n) = (Fibonacci(prime(n)) +/- 1)/prime(n), where '+' is chosen if prime(n)== 2 or 3 (mod 5), '-' is chosen otherwise. For n>=2, a(n) = round(Fibonacci(prime(n))/prime(n)). - Vladimir Shevelev, Mar 12 2014