منابع مشابه
Substitution Invariant Beatty Sequences
with θ irrational and taken to satisfy 0 < θ < 1; plainly this may be assumed without loss of generality. Evidently (fn) is a sequence of zeros and ones. Denote by w0 and w1 words on the alphabet {0, 1} ; that is, finite strings in the letters 0 and 1. Then the sequence (fn) is said to be invariant under the substitution W given by W : 0 −→ w0, 1 −→ w1, if the infinite strings fθ = f1f2f3 . . ....
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A rational Beatty sequence has the form {�pi/q + b� : i ∈ Z} where p > q > 0 and gcd(p, q) = 1. We call p/q the modulus of the sequence and b the offset. Morikawa gave a condition on the moduli of two Beatty sequences such that they would be disjoint for a suitable choice of offsets. Holzman and Fraenkel showed that the sequence formed by the intersection of two Beatty sequences with moduli p1/...
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A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form p = 2 ⌊αn⌋+1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately imply that this conjecture holds in a ...
متن کاملPrime divisors in Beatty sequences
We study the values of arithmetic functions taken on the elements of a non-homogeneous Beatty sequence αn+ β , n= 1,2, . . . , where α,β ∈R, and α > 0 is irrational. For example, we show that ∑ n N ω ( αn+ β )∼N log logN and ∑ n N (−1)Ω( αn+β ) = o(N), where Ω(k) and ω(k) denote the number of prime divisors of an integer k = 0 counted with and without multiplicities, respectively. © 2006 Elsevi...
متن کاملSumsets of Finite Beatty Sequences
An investigation of the size of S + S for a finite Beatty sequence S = (si) = (biα + γc), where b c denotes “floor”, α, γ are real with α ≥ 1, and 0 ≤ i ≤ k − 1 and k ≥ 3. For α > 2, it is shown that |S+S| depends on the number of “centres” of the Sturmian word ∆S = (si−si−1), and hence that 3(k−1) ≤ |S +S| ≤ 4k−6 if S is not an arithmetic progression. A formula is obtained for the number of ce...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1993
ISSN: 0012-365X
DOI: 10.1016/0012-365x(93)90153-k