login
Fibonacci numbers for which the product of the digits is also a Fibonacci number.
2

%I #16 Aug 27 2016 12:01:41

%S 0,1,2,3,5,8,13,21,610,10946,75025,832040,2178309,5702887,14930352,

%T 39088169,102334155,165580141,701408733,1134903170,1836311903,

%U 2971215073,4807526976,7778742049,12586269025,20365011074,32951280099

%N Fibonacci numbers for which the product of the digits is also a Fibonacci number.

%C A000045 INTERSECT A011540 is a subsequence. As a consequence of Carmichael's theorem, the product of the digits of terms in the sequence must be in the set {0, 1, 2, 3, 5, 8, 21, 144} and if a term is zeroless (A052382), then at most 6 digits are not equal to 1. Conjecture: all terms > 21 have a 0 digit, i.e. is a member of A011540. - _Chai Wah Wu_, Mar 12 2016

%H Chai Wah Wu, <a href="/A117770/b117770.txt">Table of n, a(n) for n = 1..1000</a>

%e 21 is in the sequence because (1)it is a Fibonacci number and (2)the product of its digits 2*1=2 is also a Fibonacci number.

%t With[{fibs=Fibonacci[Range[0,100]]},Select[fibs,MemberQ[fibs,Times@@ IntegerDigits[ #]]&]]//Union (* _Harvey P. Dale_, Aug 27 2016 *)

%o (Python)

%o from operator import mul

%o from functools import reduce

%o A117770_list, a, b = [0], 1, 1

%o for i in range(10**3):

%o if reduce(mul,(int(d) for d in str(b))) in (0,1,2,3,5,8,21,144):

%o A117770_list.append(b)

%o a, b = b, a+b # _Chai Wah Wu_, Mar 13 2016

%Y Cf. A000045, A117769.

%K base,nonn

%O 1,3

%A Luc Stevens (lms022(AT)yahoo.com), Apr 15 2006

%E Entries checked by _Klaus Brockhaus_, Apr 17 2006