login
a(n) = least k>0 such that n! divides Fibonacci(k).
4

%I #28 Jan 30 2014 10:46:11

%S 1,1,3,12,12,60,60,120,480,4320,43200,43200,518400,3628800,7257600,

%T 108864000,1741824000,1741824000,31352832000,31352832000,627056640000,

%U 13168189440000,289700167680000,289700167680000,6952804024320000,173820100608000000,4519322615808000000,122021710626816000000

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

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

%H Max Alekseyev, <a href="/A214528/b214528.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A001177(n!)

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

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

%o (Python)

%o n = f = c = d = 1 # f = (n-1)!

%o fc1 = fd1 = 0 # Fib[c-1], Fib[d-1]

%o fc = fd = 1 # Fib[c], Fib[d]

%o while 1:

%o if fc % f:

%o if c==d:

%o fd, fd1 = fc, fc1

%o t = fc*fc

%o fc, fc1 = (2*fc*fc1+t), (fc1*fc1+t)

%o else:

%o fc, fc1 = (fc*(fd1+fd) + fc1*fd), (fc*fd + fc1*fd1)

%o c += d

%o #print '.',

%o else:

%o print c,

%o d = c

%o f *= n

%o n += 1

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

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

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

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

%Y Cf. A000142, A000045.

%K nonn

%O 0,3

%A _Alex Ratushnyak_, Aug 08 2012

%E Terms a(17) onward from _Max Alekseyev_, Jan 30 2014