3-Smooth Representations of Integers

A Dual-Radix Modular Division Algorithm for Computing Periodic Orbits within Syracuse Dynamical Systems

“…Let f = (f w ) τ −1 w=0 and e = (e w ) τ −1 w=0 be positive partitions of order τ of F τ and E τ , respectively. For any pair (m, l) of coprime natural numbers greater than or equal to 2, we can construct a (m, l)-system that admits a sequence with the translation values a = 1 τ : we prescribe s w ≡ When the terms m = 3 and l = 2, we have a three-smooth representation ([2], [9], [4]) of the numerator of the iterate.…”

“…, τ − 1}, these rational expressions yield the iterate values in an (accelerated) orbit of length τ within the 3x+1 dynamical system (of unaccelerated length E τ ); in this context, such expressions are known as the Böhm-Sontacchi numbers [5]. The mixed-base representations of the Böhm-Sontacchi numbers are considered in [1], they have been studied in further detail (within the context of rationals with odd denominators) in [14], and the 3-smooth representations of the numerators have been studied in [2] and [4]. A notorious open question pertains to the existence of positive integers (exceeding 1) that admit such rational expressions (see [15] and [22] for comprehensive surveys on this subject).…”

On generalized Frame–Stewart numbers

“…Case 1: if p i > 1 for all 3 ≤ i ≤ k. We proceed by double induction on k ≥ 3 and n ≥ 1. For k = 3, it is clear that G 1 3 (1) = 1 and, by induction on n ≥ 1, that ∆G…”

“…These numbers are called "3-smooth numbers" [13] and have been studied extensively in number theory, in relation to the distribution of prime numbers [6] and to new number representations [1,4]. The formulation and analysis of T(n), however, has some defects such that (i) it is only focused on the 4-peg case with no consideration for the general case k ≥ 3; and (ii) even in the 4-peg case, term 2 i • α j consists of constant 2 and parameter α, which might admit further generalization.…”

The 3x+1 Problem and Integer Representations

“…Blecksmith, McCallum and Selfridge [11] consider 3-smooth numbers, which are of the form 3 k 2 a for some k, a ∈ N 0 . We credit 3-smooth numbers to Ramanujan.…”