A111366 - OEIS

%I #5 Oct 31 2013 12:17:32

%S 1,6,13,61,73,92,97,198,212,217,222,270,349,380,404,438,524,630,649,

%T 836,937,1446,1477,1513,1532,1729,2005,2046,2060,2077,2209,2348,2660,

%U 2862,2934,3265,3649,3889,4093,4609,4686,4945,5180,5444,5497,5749,5929,6102

%N Numbers such that the sum of the digits of floor(phi^n) is also the sum of the digits of the n-th Fibonacci number (in base 10), where phi is the golden ratio.

%C Questions: (1) Is this sequence infinite? (2) Are the gaps between the elements of this sequence bounded from above? (3) If this sequence is infinite, what is its asymptotic growth? (4) Consider the definition of this sequence for other values c instead of the golden ratio. What are the properties of this modified sequence?

%e trunc(phi^6) = 17, the 6th Fibonacci number is 8; the sum of their digits is the same, thus 6 is in the sequence.

%t $MaxExtraPrecision = 10^9; fQ[n_] := Plus @@ IntegerDigits@Floor@(GoldenRatio^n) == Plus @@ IntegerDigits@Fibonacci@n; Select[ Range[6108], fQ[ # ] &] (* _Robert G. Wilson v_ *)

%Y Cf. A066212, A001999.

%K base,nonn

%O 1,2

%A _Stefan Steinerberger_, Nov 07 2005

%E Edited, corrected and extended by _Robert G. Wilson v_, Nov 16 2005