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For a positive integer d, a non-negative integer n and a non-negative integer h ≤ n , we study the number C n (d) of principal ideals; and the number C n , h (d) of principal ideals generated by an element of rank h, in the d-tonal partition monoid on n elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions. [ABSTRACT FROM AUTHOR]

We study a recursively defined sequence which is constructed using the least common multiple. Several authors have conjectured that every term of that sequence is or a prime. In this paper we show that this claim is connected to a strong version of Linnik's theorem, which is still unproved. We also study a generalization that replaces the first term by any positive integer. Under this variation some composite numbers may appear now. We give a full characterization of these numbers. [ABSTRACT FROM AUTHOR]

Define to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that for all n. Define the defect of n, denoted by , to be ; in this paper we present a method for classifying all nwith for a given r. From this, we derive several consequences. We prove that for with m and k not both zero, and present a method that can, with more computation, potentially prove the same for larger m. Furthermore, defining to be the number of n with and , we prove that , allowing us to conclude that the values of can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this brief note, we formulate some modifications of the Pell-Jacobsthal numbers, which belong to a more general class, the Lucas sequences. [ABSTRACT FROM AUTHOR]