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

1,2

Links

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

Example

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

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]

Prog

(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)); }

Crossrefs

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

Subsequence of A328208 and A377209.

Sequence in context: A328208 A173643 A377209 * A120722 A090811 A162002

Adjacent sequences: A377207 A377208 A377209 * A377211 A377212 A377213

Keyword

nonn,easy,base

Author

Amiram Eldar, Oct 20 2024

Status

approved