A377209 Zeckendorf-Niven numbers (A328208) k such that k/z(k) is also a Zeckendorf-Niven number, where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k.

1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 16, 21, 24, 26, 30, 34, 36, 42, 48, 55, 60, 66, 68, 72, 78, 81, 89, 90, 108, 110, 120, 126, 135, 144, 152, 168, 178, 180, 192, 204, 207, 233, 240, 243, 264, 270, 276, 288, 300, 304, 312, 324, 330, 336, 360, 377, 380, 390, 396, 408

Offset: 1

Author

Amiram Eldar, Oct 20 2024

Keywords

• nonn
• easy
• base

Examples

			12 is a term since 12/z(12) = 4 is an integer and also 4/z(4) = 2 is an integer.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007895, A376616 (binary analog).

Subsequence of A328208.

Subsequences: A000045, A377210.

Programs

Mathematica

zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
q[k_] := Module[{z = zeck[k]}, Divisible[k, z] && Divisible[k/z, zeck[k/z]]]; Select[Range[400], q]

PARI

zeck(n) = if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s) \\ Charles R Greathouse IV at A007895
is(k) = {my(z = zeck(k)); !(k % z) && !((k/z) % zeck(k/z)); }