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A376028
Zeckendorf-Niven numbers (A328208) with a record gap to the next Zeckendorf-Niven number.
2
1, 6, 18, 30, 36, 48, 208, 5298, 6132, 6601, 8280, 12228, 17052, 68220, 113990, 120504, 438570, 1015416, 1343232, 1848400, 5338548, 12727143, 83877810, 330963120, 409185360, 418561770, 2428646640, 2834120595, 2876557200, 2940992640, 7218753758, 7306145012, 7609637140
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
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
Similar sequences: A337076, A337077, A347495, A347496, A376029.
Sequence in context: A077660 A240991 A304050 * A351220 A242394 A030568
KEYWORD
nonn,base
AUTHOR
Amiram Eldar, Sep 06 2024
STATUS
approved