A377210 Zeckendorf-Niven numbers (A328208) k such that m = k/z(k) and m/z(m) are also Zeckendorf-Niven numbers, 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, 42, 48, 55, 60, 68, 78, 89, 110, 120, 126, 144, 178, 180, 192, 204, 233, 243, 264, 270, 288, 300, 312, 324, 330, 360, 377, 466, 480, 534, 540, 576, 600, 610, 621, 672, 720, 754, 768, 864, 987, 1020, 1056

Offset: 1

Author

Amiram Eldar, Oct 20 2024

Keywords

• nonn
• easy
• base

Examples

			24 is a term since 24/z(24) = 12, 12/z(12) = 4 and 4/z(4) = 2 are all integers.
		

Links

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

Crossrefs

Cf. A000045 (a subsequence), A007895, A376617 (binary analog).

Subsequence of A328208 and A377209.

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], z2, m, n}, IntegerQ[m = k/z] && Divisible[m, z2 = zeck[m]] && Divisible[n = m/z2, zeck[n]]]; Select[Range[1000], 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), z2, m); if(k % z, return(0)); m = k/z; z2 = zeck(m); !(m % z2) && !((m/z2) % zeck(m/z2)); }