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Numbers m such that m and m+1 are consecutive Zeckendorf-Niven numbers (A328208).
27

%I #8 Dec 29 2021 05:57:37

%S 1,2,3,4,5,12,13,21,26,55,68,80,89,92,93,110,152,183,195,207,233,236,

%T 237,254,291,304,327,364,377,380,381,398,435,471,484,555,584,605,609,

%U 639,644,759,795,834,875,894,930,987,992,1004,1011,1028,1047,1076,1220

%N Numbers m such that m and m+1 are consecutive Zeckendorf-Niven numbers (A328208).

%H Amiram Eldar, <a href="/A328209/b328209.txt">Table of n, a(n) for n = 1..10000</a>

%H Helen G. Grundman, <a href="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 12 is in the sequence since both 12 and 13 are in A328208: A007895(12) = 3 is a divisor of 12, and A007895(13) = 1 is a divisor of 13.

%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, {2}]; While[c < 60, If[aQ[k], v = Join[Rest[v], {k}]; If[AllTrue[Differences[v], # == 1 &], c++; AppendTo[s, k - 1]]]; k++]; s (* after _Alonso del Arte_ at A007895 *)

%Y Cf. A005349, A007895, A328208.

%K nonn

%O 1,2

%A _Amiram Eldar_, Oct 07 2019