A339211 Zeckendorf self numbers: numbers not of the form k + A007895(k).

1, 5, 7, 10, 19, 21, 27, 29, 32, 36, 40, 42, 45, 54, 61, 63, 66, 75, 77, 83, 85, 88, 95, 97, 100, 109, 111, 117, 119, 122, 126, 130, 132, 135, 144, 146, 150, 152, 155, 164, 166, 172, 174, 177, 181, 185, 187, 190, 199, 206, 208, 211, 220, 222, 228, 230, 233, 239

Offset: 1

Author

Amiram Eldar, Nov 27 2020

Keywords

• nonn
• base

Comments

Analogous to self numbers (A003052) using the Zeckendorf representation (A014417) instead of decimal expansion.

The numbers of terms that do not exceed 10^k, for k = 0, 1, ..., are 1, 4, 25, 236, 2351, 23495, 234949, 2349463, 23494586, 234945839, 2349458364, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23494583... . - Amiram Eldar, Aug 08 2025

References

• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Self Number.
• Eric Weisstein's World of Mathematics, Zeckendorf Representation.
• Wikipedia, Self number.
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A003052, A007895, A010061, A010064, A010067, A010070, A014417, A328208, A339212, A339213, A339214, A339215.

Programs

Mathematica

z[n_] := n + Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; m = 250; Complement[Range[m], Array[z, m]] (* after Alonso del Arte at A007895 *)