A376028 Zeckendorf-Niven numbers (A328208) with a record gap to the next Zeckendorf-Niven number.

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

Author

Amiram Eldar, Sep 06 2024

Keywords

• nonn
• base

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.

Examples

			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.
		

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.

Crossrefs

Cf. A328208, A328209.

Similar sequences: A337076, A337077, A347495, A347496, A376029.

Programs

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]