A376028 - OEIS

%I #11 Sep 10 2024 08:04:59

%S 1,6,18,30,36,48,208,5298,6132,6601,8280,12228,17052,68220,113990,

%T 120504,438570,1015416,1343232,1848400,5338548,12727143,83877810,

%U 330963120,409185360,418561770,2428646640,2834120595,2876557200,2940992640,7218753758,7306145012,7609637140

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

%C The corresponding record gaps are 1, 2, 3, 4, 6, 7, 20, ... (see the link for more values).

%C Ray (2005) and Ray and Cooper (2006) proved that the asymptotic density of the Zeckendorf-Niven numbers is 0. Therefore, this sequence is infinite.

%D Andrew B. Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.

%H Amiram Eldar, <a href="/A376028/b376028.txt">Table of n, a(n) for n = 1..42</a>

%H Amiram Eldar, <a href="/A376028/a376028.txt">Table of n, a(n), gap(n) for n = 1..42</a>

%H Andrew Ray and Curtis Cooper, <a href="http://cs.ucmo.edu/~cnc8851/articles/kzeckniven.pdf">On the natural density of the k-Zeckendorf Niven numbers</a>, J. Inst. Math. Comput. Sci. Math., Vol. 19 (2006), pp. 83-98.

%e 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.

%t 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]

%Y Cf. A328208, A328209.

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

%K nonn,base

%O 1,2

%A _Amiram Eldar_, Sep 06 2024