A178763 Product of primitive prime factors of Fibonacci(n).

1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 4181, 41, 421, 199, 28657, 23, 3001, 521, 5777, 281, 514229, 31, 1346269, 2207, 19801, 3571, 141961, 107, 24157817, 9349, 135721, 2161, 165580141, 211, 433494437, 13201, 109441, 64079

Offset: 1

Author

T. D. Noe, Jun 10 2010

Keywords

• nonn

Comments

Same as A001578 for the first 18 terms.

Let b(n) be the greatest divisor of Fibonacci(n) that is coprime to Fibonacci(m) for all positive integers m < n, then a(n) = b(n) for all n, provided that no Wall-Sun-Sun prime exists. Otherwise, if p is a Wall-Sun-Sun prime and A001177(p) = k (then A001177(p^2) = k), then p^2 divides b(k), but by definition a(k) is squarefree. - Jianing Song, Jul 02 2019

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See pp. 60-64 for a table of the first 385 terms.
• Florian Luca, Carl Pomerance, and Stephen Wagner, Fibonacci Integers (preprint)

Crossrefs

Cf. A061446, A086597, A152012 (Indices of prime terms).

Programs

PARI

a(n)=my(d=divisors(n), f=fibonacci(n), t); t=lcm(apply(fibonacci,d[1..#d-1])); while((t=gcd(t,f))>1, f/=t); f \\ Charles R Greathouse IV, Nov 30 2016

Formula

a(n) = A061446(n) / A178764(n).

a(n) = A061446(n) / gcd(A061446(n), n) if n != 5, 6, provided that no Wall-Sun-Sun prime exists. - Jianing Song, Jul 02 2019