On some properties of patial sums (original) (raw)
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Sequences of integers, conjectures and new arithmetical tools
In three of my previous published books, namely “Two hundred conjectures and one hundred and fifty open problems on Fermat pseudoprimes”, “Conjectures on primes and Fermat pseudoprimes, many based on Smarandache function” and “Two hundred and thirteen conjectures on primes”, I showed my passion for conjectures on sequences of integers. In spite the fact that some mathematicians stubbornly understand mathematics as being just the science of solving and proving, my books of conjectures have been well received by many enthusiasts of elementary number theory, which gave me confidence to continue in this direction. Part One of this book brings together papers regarding conjectures on primes, twin primes, squares of primes, semiprimes, different types of pairs or triplets of primes, recurrent sequences, sequences of integers created through concatenation and other sequences of integers related to primes. Part Two of this book brings together several articles which present the notions of c-primes, m-primes, c-composites and m-composites (c/m-integers), also the notions of g-primes, s-primes, g-composites and s-composites (g/s-integers) and show some of the applications of these notions (because this is not a book structured unitary from the beginning but a book of collected papers, I defined the notions mentioned in various papers, but the best definition of them can be found in Addenda to the paper numbered tweny-nine), in the study of the squares of primes, Fermat pseudoprimes and generally in Diophantine analysis. Part Three of this book presents the notions of “Coman constants” and “Smarandache-Coman constants”, useful to highlight the periodicity of some infinite sequences of positive integers (sequences of squares, cubes, triangular numbers, polygonal numbers), respectively in the analysis of Smarandache concatenated sequences. Part Four of this book presents the notion of Smarandache-Coman sequences, id est sequences of primes formed through different arithmetical operations on the terms of Smarandache concatenated sequences. Part Five of this book presents the notion of Smarandache-Coman function, a function based on the well known Smarandache function which seems to be particularly interesting: beside other characteristics, it seems to have as values all the prime numbers and, more than that, they seem to appear, leaving aside the non-prime values, in natural order.
On certain positive integer sequences
2004
A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.
On a sum involving powers of reciprocals of an arithmetical progression
Our purpose is to establish the following result: Let a and d be coprime integers and a, a + d, a + 2d,. .. , a + (k − 1) d (k 2) be an arithmetical progression. Then for all integers α0, α1,. .. , α k−1 the rational number 1/a α 0 + 1/ (a + d) α 1 + • • • + 1/ (a + (k − 1) d) α k−1 is never an integer. This result extends theorems of Taeisinger (1915) and Kürschák (1918), and also generalizes a result of Erdős (1932).
2017
Summation of the p-adic functional series ∑ε^n n! P_k^ε (n; x) x^n , where P_k^ε (n; x) is a polynomial in x and n with rational coefficients, and ε = ± 1, is considered. The series is convergent in the domain |x|_p ≤ 1 for all primes p. It is found the general form of polynomials P_k^ε (n; x) which provide rational sums when x ∈Z. A class of generating polynomials A_k^ε (n; x) plays a central role in the summation procedure. These generating polynomials are related to many sequences of integers. This is a brief review with some new results.
2021
The purpose of this paper consists to study the sums of the type P (n) + P (n − d) + P (n − 2d) + . . . , where P is a real polynomial, d is a positive integer and the sum stops at the value of P at the smallest natural number of the form (n − kd) (k ∈ N). Precisely, for a given d, we characterize the R-vector space Ed constituting of the real polynomials P for which the above sum is polynomial in n. The case d = 2 is studied in more details. In the last part of the paper, we approach the problem through formal power series; this inspires us to generalize the spaces Ed and the underlying results. Also, it should be pointed out that the paper is motivated by the curious formula: n2 + (n− 2)2 + (n− 4)2 + · · · = n(n+1)(n+2) 6 , due to Ibn al-Banna al-Marrakushi (around 1290). MSC 2010: Primary 11B68, 11C08, 13F25.