%I #33 Jan 02 2023 12:30:50
%S 1,2,4,5,10,11,13,14,28,29,31,32,37,38,40,41,82,83,85,86,92,93,96,105,
%T 111,112,115,116,122,177,236,237,244,245,247,266,267,270,276,277,283,
%U 294,301,302,347,558,628,638,646,647,649,655,669,674,685,686
%N Indices n for which A229037(n)=1.
%C _Charles R Greathouse IV_ asked for a proof showing that this sequence is infinite (SeqFan mailing list, Jan 2014).
%C Significant jumps occur at a(3)=4=2*a(2), a(5)=10=2*a(4), a(9)=28=2*a(8), a(17)=82=2*a(16), a(31)=236 >> a(29)=122, a(47)=628 >> a(44)=302, a(70)=1622 >> a(66)=809, a(90)=4165 >> a(87)=2062, ... . Here, the size of the terms roughly doubles over the interval of very few indices. The indices such that a(n[k]) >= 2*a(n[k-1]) are n[k] = 3, 5, 9, 17, 30, 46, 69, 89, ... .
%C This sequence first differs from A003278 at the 21st term, which is 92 here but 91 in A003278. Up to 91, each natural number n that did not appear in this sequence failed to do so because there were two smaller numbers n-a and n-2a, with A229037(n-a) and A229037(n-2a) both equal to 1. 91 is missing from this sequence; in other words, A229037(91) is not 1, because A229037(27) = 9 and A229037(59) = 5. - _Jack W Grahl_, Dec 28 2014
%H Charles R Greathouse IV and Chai Wah Wu, <a href="/A236246/b236246.txt">Table of n, a(n) for n = 1..434</a>, First 266 terms from Charles R Greathouse IV.
%H Charles R Greathouse IV, <a href="http://list.seqfan.eu/oldermail/seqfan/2014-January/012322.html">[seqfan] Lexicographically first 3-free sequence</a> (2014)
%o (Haskell)
%o a236246 n = a236246_list !! (n-1)
%o a236246_list = filter ((== 1) . a229037) [1..]
%o -- _Reinhard Zumkeller_, Apr 26 2014
%o (Python)
%o A236246_list, A229037_list = [], []
%o for n in range(10**6):
%o ....i, j, b = 1, 1, set()
%o ....while n-2*i >= 0:
%o ........b.add(2*A229037_list[n-i]-A229037_list[n-2*i])
%o ........i += 1
%o ........while j in b:
%o ............b.remove(j)
%o ............j += 1
%o ....A229037_list.append(j)
%o ....if j == 1:
%o ........A236246_list.append(n+1) # _Chai Wah Wu_, Dec 25 2014
%Y Subsequence of A241673.
%K nonn
%O 1,2
%A _M. F. Hasler_, Jan 20 2014