|
|
A328208
|
|
Zeckendorf-Niven numbers: numbers divisible by the number of terms in their Zeckendorf representation (A007895).
|
|
32
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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:
|
|
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 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|