cp's OEIS Frontend

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

%I A214528 #31 Apr 08 2025 18:26:37
%S A214528 1,1,3,12,12,60,60,120,480,4320,43200,43200,518400,3628800,7257600,
%T A214528 108864000,1741824000,1741824000,31352832000,31352832000,627056640000,
%U A214528 13168189440000,289700167680000,289700167680000,6952804024320000,173820100608000000,4519322615808000000,122021710626816000000
%N A214528 a(n) = least k>0 such that n! divides Fibonacci(k).
%C A214528 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 A214528 Max Alekseyev, <a href="/A214528/b214528.txt">Table of n, a(n) for n = 0..1000</a>
%F A214528 a(n) = A001177(n!)
%e A214528 Least k such that 2! divides Fibonacci(k) is 3: Fibonacci(3)=2, so a(2)=3.
%e A214528 Least k such that 3! divides Fibonacci(k) is 12: Fibonacci(12)=144, so a(3)=12.
%o A214528 (Python)
%o A214528 n = f = c = d = 1  # f = (n-1)!
%o A214528 fc1 = fd1 = 0      # Fib[c-1], Fib[d-1]
%o A214528 fc  = fd  = 1      # Fib[c],   Fib[d]
%o A214528 while 1:
%o A214528     if fc % f:
%o A214528         if c==d:
%o A214528             fd, fd1 = fc, fc1
%o A214528             t = fc*fc
%o A214528             fc, fc1 = (2*fc*fc1+t), (fc1*fc1+t)
%o A214528         else:
%o A214528             fc, fc1 = (fc*(fd1+fd) + fc1*fd), (fc*fd + fc1*fd1)
%o A214528         c += d
%o A214528         #print('.', end=', ')
%o A214528     else:
%o A214528         print(c, end=', ')
%o A214528         d = c
%o A214528         f *= n
%o A214528         n += 1
%Y A214528 Cf. A001177 (least k such that n divides Fibonacci(k)).
%Y A214528 Cf. A132632 (least k such that n^2 divides Fibonacci(k)).
%Y A214528 Cf. A132633 (least k such that n^3 divides Fibonacci(k)).
%Y A214528 Cf. A215011 (least k such that triangular(n) divides Fibonacci(k)).
%Y A214528 Cf. A000142, A000045.
%K A214528 nonn
%O A214528 0,3
%A A214528 _Alex Ratushnyak_, Aug 08 2012
%E A214528 Terms a(17) onward from _Max Alekseyev_, Jan 30 2014