OFFSET
1,2
COMMENTS
The corresponding record gaps are 1, 2, 3, 4, 6, 7, 20, ... (see the link for more values).
Ray (2005) and Ray and Cooper (2006) proved that the asymptotic density of the Zeckendorf-Niven numbers is 0. Therefore, this sequence is infinite.
REFERENCES
Andrew B. Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..42
Amiram Eldar, Table of n, a(n), gap(n) for n = 1..42
Andrew Ray and Curtis Cooper, On the natural density of the k-Zeckendorf Niven numbers, J. Inst. Math. Comput. Sci. Math., Vol. 19 (2006), pp. 83-98.
EXAMPLE
6 is a term since it is a Zeckendorf-Niven number, and the next Zeckendorf-Niven number is 8, with a gap 8 - 6 = 2, which is a record since all the numbers below 6 are also Zeckendorf-Niven numbers.
MATHEMATICA
z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; znQ[n_] := Divisible[n, z[n]]; seq[kmax_] := Module[{gapmax = 0, gap, k1 = 1, s = {}}, Do[If[znQ[k], gap = k - k1; If[gap > gapmax, gapmax = gap; AppendTo[s, k1]]; k1 = k], {k, 2, kmax}]; s]; seq[10^4]
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Amiram Eldar, Sep 06 2024
STATUS
approved