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A052499 If n is in the sequence then so are 2n and 4n-1.

%I A052499 #40 Feb 12 2025 19:01:56
%S A052499 1,2,3,4,6,7,8,11,12,14,15,16,22,23,24,27,28,30,31,32,43,44,46,47,48,
%T A052499 54,55,56,59,60,62,63,64,86,87,88,91,92,94,95,96,107,108,110,111,112,
%U A052499 118,119,120,123,124,126,127,128,171,172,174,175,176,182,183,184,187
%N A052499 If n is in the sequence then so are 2n and 4n-1.
%C A052499 Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = 1 + A003754.
%H A052499 Reinhard Zumkeller, <a href="/A052499/b052499.txt">Table of n, a(n) for n = 0..10000</a>
%H A052499 J.-P. Allouche, J. Shallit and G. Skordev, <a href="http://dx.doi.org/10.1016/j.disc.2004.12.004">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math. 292 (2005) 1-15.
%H A052499 David Garth and Adam Gouge, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Garth/garth14.html">Affinely Self-Generating Sets and Morphisms</a>, Journal of Integer Sequences, 10 (2007) 1-13.
%H A052499 Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Hinz/hinz5.html">Precious Metal Sequences and Sierpinski-Type Graphs</a>, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
%H A052499 T. Karki, A. Lacroix, M. Rigo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Rigo/rigo6.html">On the recognizability of self-generating sets</a>, JIS 13 (2010) #10.2.2.
%H A052499 C. Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/goldentext.html">A Self-Generating Set and the Golden Mean</a>, J. Integer Sequences, 3 (2000), #00.2.8.
%F A052499 a(A001911(n)) = 2^n.
%e A052499 a(9)=14 is in the sequence because 14=2*(4*(2*1)-1).
%t A052499 1 + Select[ Range[0, 200], FreeQ[ IntegerDigits[#, 2], {___, 0, 0, ___} ] & ] (* _Jean-François Alcover_, Jan 20 2012, after J.-P. Allouche *)
%t A052499 a[1] = 1; a[n_] := a[n] = a[n - 1] + Ceiling[2^IntegerExponent[a[n - 1], 2]/3]; Array[a, 200] (* _Birkas Gyorgy_, May 30 2012 *)
%o A052499 (Haskell)
%o A052499 import Data.Set (singleton, deleteFindMin, insert)
%o A052499 a052499 n = a052499_list !! n
%o A052499 a052499_list = f $ singleton 1 where
%o A052499    f s = m : f (insert (2*m) $ insert (4*m-1) s') where
%o A052499       (m, s') = deleteFindMin s
%o A052499 -- _Reinhard Zumkeller_, Jul 06 2011
%o A052499 (Python)
%o A052499 from itertools import count, islice
%o A052499 def A052499_gen(startvalue=1): # generator of terms >= startvalue
%o A052499     return filter(lambda n: not '00' in bin(n-1),count(max(startvalue,1)))
%o A052499 A052499_list = list(islice(A052499_gen(),20)) # _Chai Wah Wu_, Feb 12 2025
%Y A052499 Cf. A001911, A003754.
%K A052499 nonn,nice
%O A052499 0,2
%A A052499 _Henry Bottomley_, Mar 15 2000