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

Author

Alex Ratushnyak, Aug 08 2012

Keywords

• nonn

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, ...

Examples

			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.
		

Links

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

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.

Programs

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('.', end=', ')
    else:
        print(c, end=', ')
        d = c
        f *= n
        n += 1

Formula

a(n) = A001177(n!)

Extensions

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