A214528 a(n) = least k>0 such that n! divides Fibonacci(k).

1, 1, 3, 12, 12, 60, 60, 120, 480, 4320, 43200, 43200, 518400, 3628800, 7257600, 108864000, 1741824000, 1741824000, 31352832000, 31352832000, 627056640000, 13168189440000, 289700167680000, 289700167680000, 6952804024320000, 173820100608000000, 4519322615808000000, 122021710626816000000

Offset

0,3

Comments

b(n) = a(n)/a(n-1) begins: 1, 3, 4, 1, 5, 1, 2, 4, 9, 10, 1, 12, 7, 2, 15, 16, ...

Links

Max Alekseyev, Table of n, a(n) for n = 0..1000

Formula

a(n) = A001177(n!)

Example

Least k such that 2! divides Fibonacci(k) is 3: Fibonacci(3)=2, so a(2)=3.
Least k such that 3! divides Fibonacci(k) is 12: Fibonacci(12)=144, so a(3)=12.

Prog

(Python)
n = f = c = d = 1  # f = (n-1)!
fc1 = fd1 = 0      # Fib[c-1], Fib[d-1]
fc  = fd  = 1      # Fib[c],   Fib[d]
while 1:
    if fc % f:
        if c==d:
            fd, fd1 = fc, fc1
            t = fc*fc
            fc, fc1 = (2*fc*fc1+t), (fc1*fc1+t)
        else:
            fc, fc1 = (fc*(fd1+fd) + fc1*fd), (fc*fd + fc1*fd1)
        c += d
        #print '.',
    else:
        print c,
        d = c
        f *= n
        n += 1

Crossrefs

Cf. A001177 (least k such that n divides Fibonacci(k)).

Cf. A132632 (least k such that n^2 divides Fibonacci(k)).

Cf. A132633 (least k such that n^3 divides Fibonacci(k)).

Cf. A215011 (least k such that triangular(n) divides Fibonacci(k)).

Cf. A000142, A000045.

Sequence in context: A340354 A263672 A233287 * A217763 A234749 A024546

Adjacent sequences: A214525 A214526 A214527 * A214529 A214530 A214531

Keyword

nonn

Author

Alex Ratushnyak, Aug 08 2012

Extensions

Terms a(17) onward from Max Alekseyev, Jan 30 2014

Status

approved