A339212 - OEIS

%I #8 Nov 29 2020 02:10:08

%S 1,4,8,10,14,17,19,28,31,33,39,41,50,53,55,59,63,66,68,74,76,85,88,90,

%T 97,106,109,111,115,119,122,124,130,132,141,144,146,153,156,158,164,

%U 166,175,178,180,187,196,199,201,205,209,212,214,220,222,231,234,236

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

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

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

%H Amiram Eldar, <a href="/A339212/b339212.txt">Table of n, a(n) for n = 1..10000</a>

%H J. L. Brown, Jr., <a href="http://www.fq.math.ca/Scanned/3-1/brown.pdf">A new characterization of the Fibonacci numbers</a>, Fibonacci Quarterly, Vol. 3, No. 1 (1965) pp. 1-8.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SelfNumber.html">Self Number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Self_number">Self number</a>.

%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>

%t 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]]

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

%K nonn,base

%O 1,2

%A _Amiram Eldar_, Nov 27 2020