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A099267 Numbers generated by the golden sieve.

%I A099267 #48 Mar 29 2025 12:17:45
%S A099267 2,3,5,6,8,10,11,13,14,16,18,19,21,23,24,26,27,29,31,32,34,35,37,39,
%T A099267 40,42,44,45,47,48,50,52,53,55,57,58,60,61,63,65,66,68,69,71,73,74,76,
%U A099267 78,79,81,82,84,86,87,89,90,92,94,95,97,99,100,102,103,105,107,108,110
%N A099267 Numbers generated by the golden sieve.
%C A099267 Let f(n) denote the n-th term of the current working sequence. Start with the positive integers:
%C A099267 1,2,3,4,5,6,7,8,9,10,11,12,...
%C A099267 Delete the term in position f(1), which is f(f(1))=f(1)=1, leaving:
%C A099267 2,3,4,5,6,7,8,9,10,11,12,...
%C A099267 Delete the term in position f(2), which is f(f(2))=f(3)=4, leaving:
%C A099267 2,3,5,6,7,8,9,10,11,12,...
%C A099267 Delete the term in position f(3), which is f(f(3))=f(5)=7, leaving:
%C A099267 2,3,5,6,8,9,10,11,12,...
%C A099267 Delete the term in position f(4), which is f(f(4))=f(6)=9, leaving:
%C A099267 2,3,5,6,8,10,11,12,...
%C A099267 Iterating the "sieve" indefinitely produces the sequence:
%C A099267 2,3,5,6,8,10,11,13,14,16,18,19,21,23,24,26,27,29,31,32,34,35,37,39,...
%C A099267 Positions of 1 in A189479. - _Clark Kimberling_, Apr 22 2011
%H A099267 Reinhard Zumkeller, <a href="/A099267/b099267.txt">Table of n, a(n) for n = 1..10000</a>
%H A099267 Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, <a href="https://arxiv.org/abs/2503.19696">Fibonacci-like partitions and their associated piecewise-defined permutations</a>, arXiv:2503.19696 [math.CO], 2025. See p. 3.
%H A099267 <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F A099267 a(n) = floor(n*phi + 2 - phi) where phi = (1 + sqrt(5))/2.
%F A099267 a(a(...a(1)...)) with n iterations equals F(n+1) = A000045(n+1).
%F A099267 For n>0 and k>0 we have a(a(n) + F(k) - (1 + (-1)^k)/2) = a(a(n)) + F(k+1) - 1 - (-1)^k. - _Benoit Cloitre_, Nov 22 2004
%F A099267 a(n) = a(a(n)) - n. - _Marc Morgenegg_, Sep 23 2019
%t A099267 t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0, 1}}] &, {0}, 6] (*A189479*)
%t A099267 Flatten[Position[t, 0]] (*A007066*)
%t A099267 Flatten[Position[t, 1]] (*A099267*)
%o A099267 (Haskell)
%o A099267 a099267 n = a099267_list !! (n-1)
%o A099267 a099267_list = f 1 [1..] 0 where
%o A099267    f k xs y = ys' ++ f (k+1) (ys ++ xs') g where
%o A099267      ys' = dropWhile (< y) ys
%o A099267      (ys,_:xs') = span (< g) xs
%o A099267      g = xs !! (h - 1)
%o A099267      h = xs !! (k - 1)
%o A099267 -- _Reinhard Zumkeller_, Sep 18 2011
%Y A099267 Numbers n such that a(n+1)-a(n)=2 are given by A004956.
%Y A099267 If prefixed by an initial 1, same as A026355.
%Y A099267 Cf. A001622, A136119, A007066, A189479.
%Y A099267 Complement of A007066. - _Gerald Hillier_, Dec 19 2008
%Y A099267 Cf. A193213 (primes).
%K A099267 nonn,easy,nice
%O A099267 1,1
%A A099267 _Benoit Cloitre_, Nov 15 2002