A339212 Dual-Zeckendorf self numbers: numbers not of the form k + A112310(k).

1, 4, 8, 10, 14, 17, 19, 28, 31, 33, 39, 41, 50, 53, 55, 59, 63, 66, 68, 74, 76, 85, 88, 90, 97, 106, 109, 111, 115, 119, 122, 124, 130, 132, 141, 144, 146, 153, 156, 158, 164, 166, 175, 178, 180, 187, 196, 199, 201, 205, 209, 212, 214, 220, 222, 231, 234, 236

Offset: 1

Author

Amiram Eldar, Nov 27 2020

Keywords

• nonn
• base

Comments

Analogous to self numbers (A003052) using the dual Zeckendorf representation (A104326) instead of decimal expansion.

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
• J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly, Vol. 3, No. 1 (1965) pp. 1-8.
• Eric Weisstein's World of Mathematics, Self Number.
• Wikipedia, Self number.
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A003052, A010061, A010064, A010067, A010070, A104326, A112310, A328212, A339211, A339213, A339214, A339215.

Programs

Mathematica

fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr]; dzs[n_] := n + Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]]; m = 240; Complement[Range[m], Array[dzs, m]]