A111366 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.

1, 6, 13, 61, 73, 92, 97, 198, 212, 217, 222, 270, 349, 380, 404, 438, 524, 630, 649, 836, 937, 1446, 1477, 1513, 1532, 1729, 2005, 2046, 2060, 2077, 2209, 2348, 2660, 2862, 2934, 3265, 3649, 3889, 4093, 4609, 4686, 4945, 5180, 5444, 5497, 5749, 5929, 6102

Offset: 1

Author

Stefan Steinerberger, Nov 07 2005

Keywords

• base
• nonn

Comments

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?

Examples

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

Crossrefs

Cf. A066212, A001999.

Programs

Mathematica

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

Extensions

Edited, corrected and extended by Robert G. Wilson v, Nov 16 2005