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

1,4

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Wikipedia, Prime divisors of Fibonacci numbers

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

Example

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

Mathematica

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

Crossrefs

Cf. A000045, A092330.

Sequence in context: A074128 A061226 A134827 * A134789 A058217 A306616

Adjacent sequences: A222358 A222359 A222360 * A222362 A222363 A222364

Keyword

nonn

Author

Jonathan Sondow, Feb 23 2013

Status

approved