The Ulam Sequence

Problem's Description

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).

1. Article A Hidden Signal in the Ulam Sequence

   • Stefan Steinerberger

   Experimental Mathematics 23, 460–467, 2017arXiv

2. Article The Unreasonable Rigidity of Ulam Sets

   • Joshua Hinman,
   • Borys Kuca,
   • Alexander Schlesinger,
   • Arseniy Sheydvasser

   Journal of Number TheoryarXiv

3. Book Problems in Modern Mathematics

   • Stanislaw Ulam

   year of publication: 1964

4. Other An Efficient Method for Computing Ulam Numbers

   • Philip Gibbs

   fulltext

5. Other The Ulam Numbers up to One Trillion

   • Philip Gibbs,
   • Judson McCranie

   fulltext