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A330712 Numbers k such that F(k) - 1 is divisible by floor((k - 1)/2), where F(k) is the k-th Fibonacci number (A000045).

%I A330712 #13 Dec 30 2019 12:18:20
%S A330712 3,4,5,7,15,22,25,26,27,35,41,47,49,50,73,74,75,87,89,95,97,98,101,
%T A330712 107,121,122,135,145,146,147,167,193,194,195,207,215,217,218,221,227,
%U A330712 241,242,255,275,289,290,315,327,335,337,338,347,361,362,385,386,387,395
%N A330712 Numbers k such that F(k) - 1 is divisible by floor((k - 1)/2), where F(k) is the k-th Fibonacci number (A000045).
%C A330712 Numbers of the form F(k) - 1 have the same Zeckendorf (A014417) and dual Zeckendorf (A104326) representations: alternating digits of 1 and 0 whose sum is floor((k - 1)/2). Thus, if k is in this sequence then F(k) - 1 is both a Zeckendorf-Niven number (A328208) and a lazy-Fibonacci-Niven number (A328212), i.e., A000071(a(n)) is in A330711.
%H A330712 Amiram Eldar, <a href="/A330712/b330712.txt">Table of n, a(n) for n = 1..10000</a>
%e A330712 7 is in this sequence since F(7) - 1 = 13 - 1 = 12 is divisible by floor((7 - 1)/2) = 3. The Zeckendorf and dual Zeckendorf representations of 7 are both 1010, whose sum of digits, 2, divides 12. Thus 12 is both a Zeckendorf-Niven number and a lazy-Fibonacci-Niven number.
%t A330712 Select[Range[3, 400], Divisible[Fibonacci[#] - 1, Floor[(# - 1)/2]] &]
%Y A330712 Cf. A000045, A000071, A328208, A328212, A330711.
%K A330712 nonn
%O A330712 1,1
%A A330712 _Amiram Eldar_, Dec 27 2019