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%I #10 Jan 05 2025 19:51:41
%S 1,2,3,123543,124242,545502,1367583,1856349,2431230,2465110,2593590,
%T 2783709,3247389,3479229,3917823,3942909,4174749,4303428,4494390,
%U 4920640,5143830,5710383,6261309,6493149,6552903,6956829,7420509,7470880,8970948,9107790,9507069,10952928
%N Starts of runs of 4 consecutive Zeckendorf-Niven numbers (A328208).
%C 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.
%C She also proved that the only starts of runs of 5 consecutive Zeckendorf-Niven numbers are 1 and 2.
%H Amiram Eldar, <a href="/A328211/b328211.txt">Table of n, a(n) for n = 1..216</a>
%H Helen G. Grundman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/45-3/grundman.pdf">Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers</a>, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.
%e 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.
%t 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 *)
%Y Cf. A005349, A007895, A141769, A328208.
%K nonn
%O 1,2
%A _Amiram Eldar_, Oct 07 2019