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A328208 Zeckendorf-Niven numbers: numbers divisible by the number of terms in their Zeckendorf representation (A007895).

%I A328208 #9 Jan 05 2025 19:51:41
%S A328208 1,2,3,4,5,6,8,10,12,13,14,16,18,21,22,24,26,27,30,34,36,42,45,48,55,
%T A328208 56,58,60,66,68,69,72,76,78,80,81,84,89,90,92,93,94,96,99,102,105,108,
%U A328208 110,111,116,120,126,132,135,140,144,146,150,152,153,156,159,162
%N A328208 Zeckendorf-Niven numbers: numbers divisible by the number of terms in their Zeckendorf representation (A007895).
%D A328208 Andrew Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.
%H A328208 Robert Israel, <a href="/A328208/b328208.txt">Table of n, a(n) for n = 1..10000</a>
%H A328208 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.
%H A328208 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 A328208 12 is in the sequence since A007895(12) = 3 and 3 is a divisor of 12.
%p A328208 fib:= combinat:-fibonacci:
%p A328208 phi:= 1/2 + sqrt(5)/2:
%p A328208 fibapp:= n -> phi^n/sqrt(5):
%p A328208 invfib := proc(x::posint)
%p A328208   local q, n;
%p A328208   q:= evalf((ln(x+1/2) + ln(5)/2)/ln(phi));
%p A328208   n:= floor(q);
%p A328208   if fib(n) <= x then
%p A328208     while fib(n+1) <= x do
%p A328208       n := n+1
%p A328208     end do
%p A328208   else
%p A328208     while fib(n) > x do
%p A328208       n := n-1
%p A328208     end do
%p A328208   end if;
%p A328208   n
%p A328208 end proc:
%p A328208 zeck:= proc(x) local n;
%p A328208  if x = 0 then 0
%p A328208  else
%p A328208    n:= invfib(x);
%p A328208    F[n] + zeck(x-fib(n));
%p A328208  fi
%p A328208 end proc:
%p A328208 filter:= n -> n mod nops(zeck(n)) = 0:
%p A328208 select(filter, [$1..200]); # _Robert Israel_, Oct 25 2019
%t A328208 z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; aQ[n_] := Divisible[n, z[n]]; Select[Range[1000], aQ] (* after _Alonso del Arte_ at A007895 *)
%Y A328208 Cf. A005349, A007895.
%K A328208 nonn
%O A328208 1,2
%A A328208 _Amiram Eldar_, Oct 07 2019