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

1,2

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

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]]

Crossrefs

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

Sequence in context: A320497 A310988 A161667 * A063087 A310989 A310990

Adjacent sequences: A339209 A339210 A339211 * A339213 A339214 A339215

Keyword

nonn,base

Author

Amiram Eldar, Nov 27 2020

Status

approved