A328211 Starts of runs of 4 consecutive Zeckendorf-Niven numbers (A328208).

1, 2, 3, 123543, 124242, 545502, 1367583, 1856349, 2431230, 2465110, 2593590, 2783709, 3247389, 3479229, 3917823, 3942909, 4174749, 4303428, 4494390, 4920640, 5143830, 5710383, 6261309, 6493149, 6552903, 6956829, 7420509, 7470880, 8970948, 9107790, 9507069, 10952928

Offset

1,2

Comments

Grundman proved that this sequence is infinite by showing the F(120k-6) + F(8) + F(6) + F(4) is a term for all k >= 1, where F(k) is the k-th Fibonacci number.

She also proved that the only starts of runs of 5 consecutive Zeckendorf-Niven numbers are 1 and 2.

Links

Amiram Eldar, Table of n, a(n) for n = 1..216

Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.

Example

1 is in the sequence since 1, 2, 3 and 4 are in A328208: A007895(1) = 1 is a divisor of 1, A007895(2) = 1 is a divisor of 2, A007895(3) = 1 is a divisor of 3, and A007895(4) = 2 is a divisor of 4.

Mathematica

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; aQ[n_] := Divisible[n, z[n]]; c = 0; k = 1; s = {}; v = Table[-1, {4}]; While[c < 32, If[aQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 3]]]; k++]; s (* after Alonso del Arte at A007895 *)

Crossrefs

Cf. A005349, A007895, A141769, A328208.

Sequence in context: A216977 A331426 A300191 * A324154 A038537 A338404

Adjacent sequences: A328208 A328209 A328210 * A328212 A328213 A328214

Keyword

nonn

Author

Amiram Eldar, Oct 07 2019

Status

approved