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

1,2

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.

Kelly Lao, Steven J. Miller, Nicholas Rosa, Mark Shiliaev, Garrett Tresch, Tony W. H. Wong, and Han Zhang, On Zeckendorf-Niven numbers and arithmetic progressions, arXiv preprint (2026). arXiv:2606.24006 [math.NT]

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.

Example

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

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 *)

Prog

(PARI) is(n)=my(m=n); if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=m, ); while(k && m, t=fibonacci(k); if(t<=m, m-=t; s++); k--); n%s==0) \\ Charles R Greathouse IV, Jun 24 2026

Crossrefs

Cf. A005349, A007895.

Sequence in context: A005423 A067319 A086049 * A173643 A377209 A377210

Adjacent sequences: A328205 A328206 A328207 * A328209 A328210 A328211

Keyword

nonn,changed

Author

Amiram Eldar, Oct 07 2019

Status

approved