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Leonetti and Luca ['On the iterates of the shifted Euler's function', Bull. Aust. Math. Soc. , to appear] have shown that the integer sequence $(x_n)_{n\geq 1}$ defined by $x_{n+2}=\phi (x_{n+1})+\phi (x_{n})+k$ , where $x_1,x_2\geq 1$ , $k\geq 0$ and $2 \mid k$ , is bounded by $4^{X^{3^{k+1}}}$ , where $X=(3x_1+5x_2+7k)/2$. We improve this result by showing that the sequence $(x_n)$ is bounded by $2^{2X^2+X-3}$ , where $X=x_1+x_2+2k$. [ABSTRACT FROM AUTHOR]

In this paper, the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function x x , establish two closed-form and explicit formulas for partial Bell polynomials at some specific arguments, and present several new closed-form and explicit formulas for the nth derivative of the power-exponential function x x and for related functions and integer sequences. [ABSTRACT FROM AUTHOR]

Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra A. In this setting, we prove that the indecomposable uniserial A -modules with a mast of multiplicity one in each vertex are Zavadskij modules. Since the Zavadskij property carries over to direct summands, but it is not invariant under the formation of direct sums, we give a criterion to determine when the direct sum of indecomposable Zavadskij modules is again a Zavadskij module. In addition, we use the triangulations of the n + 3 -gon associated with the cluster-tilted algebra of type A n to give a formula for the number of indecomposable Zavadskij modules over any cluster-tilted algebra of type A n . In this case, the formula gives the dimension of the cluster-tilted algebra. As an application, we discuss some integer sequences in the OEIS (The On-Line Encyclopedia of Integer Sequences) that allow us to enumerate indecomposable Zavadskij modules. [ABSTRACT FROM AUTHOR]

The four subspace problem is a known matrix problem, which is equivalent to determining all the indecomposable representations of a poset consisting of four incomparable points. In this paper, we use solutions of this problem and invariants associated with indecomposable projective modules with some suitable Brauer configuration algebras to categorify the integer sequence encoded in the OEIS as A100705 and some related integer sequences. [ABSTRACT FROM AUTHOR]

Bijections between invariants associated with indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated with solutions of the Kronecker problem are used to categorify integer sequences in the sense of Ringel and Fahr. Dimensions of the Brauer configuration algebras and their corresponding centers involved in the different processes are also given. [ABSTRACT FROM AUTHOR]

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]

We have exhaustively enumerated all simple, connected graphs of a finite order and have computed a selection of invariants over this set. Integer sequences were constructed from these invariants and checked against the Online Encyclopedia of Integer Sequences (OEIS). 141 new sequences were added and six sequences were extended. From the graph database, we were able to programmatically suggest relationships among the invariants. It will be shown that we can readily visualize any sequence of graphs with a given criteria. The code has been released as an open-source framework for further analysis and the database was constructed to be extensible to invariants not considered in this work. [ABSTRACT FROM AUTHOR]

Text The sequence of middle divisors is shown to be unbounded. For a given number n , a n , 0 is the number of divisors of n between n / 2 and 2 n . We explicitly construct a sequence of numbers n ( i ) and a list of divisors in the interesting range, so that the length of the list goes to infinity as i increases. Video For a video summary of this paper, please visit https://youtu.be/drODtRj0gjM . [ABSTRACT FROM AUTHOR]

We find the asymptotic behaviour of a sequence considered by Shaun Cooper. [ABSTRACT FROM AUTHOR]

For a finite set of points X on the unit hypersphere in ℝd we consider the iteration ui+1 = ui + χi, where χi is the point of X farthest from ui. Restricting to the case where the origin is contained in the convex hull of X we study the maximal length of ui. We give sharp upper bounds for the length of ui independently of X. Precisely, this upper bound is infinity for d ≥ 3 and for d = 2. [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]

The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope and respectively. We study extensions to this game where a player may also remove simultaneously tokens from either of the piles and from the other, where are given positive integers and where t ranges over the positive integers. We prove that for certain pairs ( p, q) the P-positions are identical to those of Wythoff Nim, but for they do not even lie on two beams. By several experimental results, we conjecture a classification of all pairs ( p, q) for which Wythoff Nim's beams of P-positions transform via a certain splitting behavior, similar to that of going from 2-pile Nim to Wythoff Nim. [ABSTRACT FROM AUTHOR]

Abstract: In this study, we compute the spectral norm of the matrix , where are ith components of any integer sequence . Also, we give a corollary and examples related to spectral norm. [Copyright &y& Elsevier]

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]