Let L0 = 2, L1 = 1, and Ln = Ln−1+Ln−2 for n ≥ 2, denote the sequence L of Lucas numbers. For any modulus m ≥ 2, and residue b (mod m), denote by vL(m, b) the number of occurrences of b as a residue in one (shortest) period of L modulo m. In this paper, we completely describe the functions vL(p , .) for k ≥ 1 in the cases p = 2 and p = 5. Using a notion formally introduced by Carlip and Jacobso...

for the arithmetic mean of the distinct prime factors of n, then pa(n) is uniformly distributed modulo 1. Furthermore, it was also shown that for large x the number of n ≤ x such that pa(n) is an integer of order of magnitude O(x/ log log x). This was made more precise by the first author in [4], who showed that the counting function of the above n is asymptotically equal to (c+o(1))x/ log log ...

Abstract. We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GLpNq Maass cusp forms for all N > 2, satisfy a central limit theorem in a suitable range, generalizing the case N “ 2 treated by É. Fouvry, S. Ganguly, E. Kowalski and P. Michel in [4]. Such universal Gaussian behaviour relies on a deep e...

Wemodify the recent method of J.-M. Deshouillers and H. Iwaniec in the theory of uniform distribution to show that the sequence with general term an = 1 n ∑ m≤n σ(m) is uniformly distributed modulo 1. We also study uniform distribution modulo 1 of some sequences related by the functions σ and φ. AMS subject classifications: 11K65, 11K31, 11J71, 11N36, 11N64

Let F0 = 0, F1 = 1, and Fn = Fn−1 + Fn−2 for n ≥ 2 denote the sequence F of Fibonacci numbers. For any modulus m ≥ 2 and residue b (modm), denote by vF (m, b) the number of occurrences of b as a residue in one (shortest) period of F modulo m. Moreover, let vL(m, b) be similarly defined for the Lucas sequence L satisfying L0 = 2, L1 = 1, and Ln = Ln−1 + Ln−2 for n ≥ 2. In this paper, completing ...

The idea of a retroactive version of a data structure is to maintain a time-ordered sequence of operations while allowing the user to revise the operation sequence by invoking and revoking (i.e., inserting and deleting, respectively) operations anywhere in the sequence—including backwards in time. In many applications of retroactivity, operations depend on the outcomes of previous queries, and ...

At present, substation operation ticket system is developed based on an expert system, which has some problems such as knowledge base redundancy, intelligence deficiency and automatic learning ability. To solve this problem, paper proposes sequence reasoning model the of Neo4j graph DuelingDQN (Dueling Deep Q Network) algorithm. Firstly, diagram structure main wiring was established using graph...