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%I #21 Oct 30 2020 15:04:54
%S 3,5,9,10,15,17,18,27,33,34,36,45,51,54,63,65,66,68,85,99,102,119,129,
%T 130,132,136,153,165,170,187,195,198,204,221,231,238,255,257,258,260,
%U 264,297,325,330,363,387,390,396,429,455,462,495,513,514,516,520,528
%N Those positive integers n such that n = (2^m +1)*k, each for some positive integer m, and k < 2^m.
%C Written in binary, each term consists of a given series of digits repeated twice, once at the beginning of the number and once at the end, separated by any number of 0's.
%H Robert Israel, <a href="/A175468/b175468.txt">Table of n, a(n) for n = 1..10000</a>
%H Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran, and Jeffrey Shallit, <a href="https://arxiv.org/abs/1710.04247">Lagrange's Theorem for Binary Squares</a>, arXiv:1710.04247 [math.NT], 2017-2018.
%H Aayush Rajasekaran, <a href="https://uwspace.uwaterloo.ca/bitstream/handle/10012/13202/Rajasekaran_Aaayush.pdf?sequence=3">Using Automata Theory to Solve Problems in Additive Number Theory</a>, MS thesis, University of Waterloo, 2018.
%e The first few terms written in binary: 11, 101, 1001, 1010, 1111, 10001, 10010, 11011. For instance, a(7) = 18 is 10010 in binary. This binary representation is made up of a 10 (2 in decimal) occurring both at the beginning and the end, with a single 0 between.
%p N:= 1000: # to get all terms <= N
%p A:= {seq(seq((2^m+1)*k, k=1..min(2^m-1,floor(N/(2^m+1)))),m=1..ilog2(N-1))}:
%p sort(convert(A,list)); # _Robert Israel_, Feb 08 2016
%t With[{n = 528}, Union@ Flatten@ Table[(2^m + 1) k, {m, Floor@ Log2[n - 1]}, {k, Min[2^m - 1, Floor[n/(2^m + 1)]]}]] (* _Michael De Vlieger_, Mar 14 2018 *)
%Y Cf. A175469, A175470.
%K nonn
%O 1,1
%A _Leroy Quet_, May 24 2010
%E More terms from _Jon E. Schoenfield_, Jun 13 2010
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