2015/07/01 by Steinerberger, Stefan · 1 citation
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1507.00267
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: n∈ ℕ\ mod~2π generates an absolutely continuous non-uniform measure supported on a subset of \mathbbT. Indeed, for the first 107 elements of Ulam's sequence, cos( 2.5714474995~ an) lt; 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.