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

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, 56, 58, 60, 66, 68, 69, 72, 76, 78, 80, 81, 84, 89, 90, 92, 93, 94, 96, 99, 102, 105, 108, 110, 111, 116, 120, 126, 132, 135, 140, 144, 146, 150, 152, 153, 156, 159, 162

Offset: 1

Author

Amiram Eldar, Oct 07 2019

Keywords

• nonn

Examples

			12 is in the sequence since A007895(12) = 3 and 3 is a divisor of 12.
		

References

• Andrew Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.
• Andrew Ray and Curtis Cooper, On the natural density of the k-Zeckendorf Niven numbers, J. Inst. Math. Comput. Sci. Math., Vol. 19 (2006), pp. 83-98.

Crossrefs

Cf. A005349, A007895.

Programs

Maple

fib:= combinat:-fibonacci:
phi:= 1/2 + sqrt(5)/2:
fibapp:= n -> phi^n/sqrt(5):
invfib := proc(x::posint)
  local q, n;
  q:= evalf((ln(x+1/2) + ln(5)/2)/ln(phi));
  n:= floor(q);
  if fib(n) <= x then
    while fib(n+1) <= x do
      n := n+1
    end do
  else
    while fib(n) > x do
      n := n-1
    end do
  end if;
  n
end proc:
zeck:= proc(x) local n;
 if x = 0 then 0
 else
   n:= invfib(x);
   F[n] + zeck(x-fib(n));
 fi
end proc:
filter:= n -> n mod nops(zeck(n)) = 0:
select(filter, [$1..200]); # Robert Israel, Oct 25 2019

Mathematica

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 *)