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

1,1

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

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.

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

Cf. A175469, A175470.

Sequence in context: A230385 A269399 A345916 * A286065 A316296 A344293

Adjacent sequences: A175465 A175466 A175467 * A175469 A175470 A175471

Keyword

nonn

Author

Leroy Quet, May 24 2010

Extensions

More terms from Jon E. Schoenfield, Jun 13 2010

Status

approved