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A175468
Those positive integers n such that n = (2^m +1)*k, each for some positive integer m, and k < 2^m.
6
3, 5, 9, 10, 15, 17, 18, 27, 33, 34, 36, 45, 51, 54, 63, 65, 66, 68, 85, 99, 102, 119, 129, 130, 132, 136, 153, 165, 170, 187, 195, 198, 204, 221, 231, 238, 255, 257, 258, 260, 264, 297, 325, 330, 363, 387, 390, 396, 429, 455, 462, 495, 513, 514, 516, 520, 528
OFFSET
1,1
COMMENTS
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.
LINKS
Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran, and Jeffrey Shallit, Lagrange's Theorem for Binary Squares, arXiv:1710.04247 [math.NT], 2017-2018.
Aayush Rajasekaran, Using Automata Theory to Solve Problems in Additive Number Theory, MS thesis, University of Waterloo, 2018.
EXAMPLE
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.
MAPLE
N:= 1000: # to get all terms <= N
A:= {seq(seq((2^m+1)*k, k=1..min(2^m-1, floor(N/(2^m+1)))), m=1..ilog2(N-1))}:
sort(convert(A, list)); # Robert Israel, Feb 08 2016
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A230385 A269399 A345916 * A286065 A316296 A344293
KEYWORD
nonn
AUTHOR
Leroy Quet, May 24 2010
EXTENSIONS
More terms from Jon E. Schoenfield, Jun 13 2010
STATUS
approved