Stanislaw Ulam, in a 1964 book, described what is now called the "Ulam sequence". It is defined by setting a_1 = 1, a_2 =2 and then picking a_n in a greedy manner as the smallest integer that can be uniquely written as the sum of two distinct earlier elements of the sequence. This results in the sequence

1,2,3,4,6,8,11, ...

The main problem is the following: there seems to exist a real number x (with x~2.571447) such that sequence a_n*x has a very strange distribution modulo 2*pi (more precisely, it seems to have an absolutely continuous distribution function that is compactly supported on a subinterval, we refer to the references). It has become clear that this is not an isolated phenomenon and that there are several related sequences that exhibit the same phenomenon. The problem is completely open. Even more basic questions are wide open (such as: does the sequence grow linearly? the best known bound is a_n < c^n for c being the golden ratio).

If you plan to formulate more than one problem all sharing the same background (e.g. they are all from the same paper) then please choose "Group", otherwise select "Single" option.

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