The Büchi sequences and Hilbert's Tenth Problem
In this short survey paper we state the Büchi conjecture and discuss its relations with the Hilbert Tenth Problem. We give some generalizations of the conjecture, and include some numerical examples.
The Collatz problem and analogues.
The cube recurrence.
The distribution of trees.
The EKG sequence.
The Erdős-Ginzburg-Ziv theorem in Abelian non-cyclic groups.
The Fibonacci number of a grid graph and a new class of integer sequences.
The fractional part of and Beatty sequences
The Haar measure of certain sets in the Bohr group
The Hankel transform and some of its properties.
The integer sequence A002620 and upper antagonistic functions.
The joint distribution of the binary digits of integer multiples
The largest subset in [1,n] whose integers have pairwise l.c.m. not exceeding n, II
The number of squares and sets
The propinquity of divisors
The restricted Toda chain, exponential Riordan arrays, and Hankel transforms.
The semigroup of nonempty finite subsets of rationals.
The sequence of fractional parts of roots
We study the function , where θ is a positive real number, ⌊·⌋ and · are the floor and fractional part functions, respectively. Nathanson proved, among other properties of , that if log θ is rational, then for all but finitely many positive integers n, . We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy . Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued...
The set of rational cycles for the 3x+1 problem
Theoretical and computational bounds for m-cycles of the 3n+1-problem