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A274695 a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that a(1)*a(2)*...*a(n) + 1 is a Fibonacci number.

%I A274695 #16 Mar 14 2020 17:21:29
%S A274695 1,2,6,133,97479304649455554938377
%N A274695 a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that a(1)*a(2)*...*a(n) + 1 is a Fibonacci number.
%C A274695 a(6) = (Fibonacci(7937)-1)/(a(2)*a(3)*a(4)*a(5)) has 1633 digits and it is thus too large to be included in Data section or in a b-file. - _Giovanni Resta_, Jul 05 2016
%e A274695 After a(1)=1 and a(2)=2, we want m, the smallest number > 2 such that 1*2*m+1 is a Fibonacci number: this is m = 6 = a(3).
%t A274695 a[1] = 1; a[n_] := a[n] = Block[{p = Times @@ Array[a, n-1], i, m}, For[i=2, ! (IntegerQ[m = (Fibonacci[i] - 1)/p] && m > a[n-1]), i++]; m]; Array[a, 6] (* _Giovanni Resta_, Jul 05 2016 *)
%o A274695 (Sage)
%o A274695 product = 1
%o A274695 seq = [ product ]
%o A274695 prev_fib_index = 0
%o A274695 max_n = 5
%o A274695 for n in range(2, max_n+1):
%o A274695     fib_index = prev_fib_index + 1
%o A274695     found = False
%o A274695     while not found:
%o A274695         fib_minus_1 = fibonacci(fib_index) - 1
%o A274695         if product.divides(fib_minus_1):
%o A274695             m = int( fib_minus_1 / product )
%o A274695             if m > seq[-1]:
%o A274695                 product = product * m
%o A274695                 seq.append( m )
%o A274695                 found = True
%o A274695                 prev_fib_index = fib_index
%o A274695                 break
%o A274695         fib_index += 1
%o A274695 print(seq)
%Y A274695 Cf. A000045, A046966.
%K A274695 nonn
%O A274695 1,2
%A A274695 _Robert C. Lyons_, Jul 04 2016
%E A274695 a(5) from _Giovanni Resta_, Jul 05 2016