A Hidden Signal in the Ulam Sequence

The Ulam sequence is defined as a1 = 1, a2 = 2 and an being the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, . . . Ulam remarked that understanding the sequence, which has been described as ’quite erratic’, seems difficult and indeed nothing is known. We report the empirical discovery of a surprising global rigidity phenomenon: there seems to exist a real α ∼ 2.5714474995 . . . such that {αan mod 2π : n ∈ N} generates an absolutely continuous non-uniform measure supported on a subset of T. Indeed, for the first 107 elements of Ulam’s sequence, cos (2.5714474995 an) < 0 for all an / ∈ {2, 3, 47, 69} . The same phenomenon arises for some other initial conditions a1, a2: the distribution functions look very different from each other and have curious shapes. A similar but more subtle phenomenon seems to arise in Lagarias’ variant of MacMahon’s ’primes of measurement’ sequence.