A175468 Those positive integers n such that n = (2^m +1)*k, each for some positive integer m, and k < 2^m.
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
Keywords
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- 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.
Programs
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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 *)
Extensions
More terms from Jon E. Schoenfield, Jun 13 2010
Comments