Results 1 to 10 of about 31,249 (173)
On a Sum Involving the Sum-of-Divisors Function [PDF]
Journal of Mathematics, 2021 Let σn be the sum of all divisors of n and let t be the integral part of t. In this paper, we shall prove that ∑n≤xσx/n=π2/6x log x+Oxlog x2/3log2 x4/3 for x⟶∞, and that the error term of this asymptotic formula is Ωx.
Feng Zhao, Jie Wu
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Iterating the Sum-of-Divisors Function [PDF]
Experimental Mathematics, 1996 Let $\sigma^0(n) = n$ and $\sigma^m(n) = \sigma(\sigma^{m-1}(n))$, where $m\ge1$ and $\sigma$ is the sum-of-divisors function. We say that $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$. We have tabulated all $(2,k)$-perfect numbers up to $10^9$ and all $(3,k)$- and $(4,k)$-perfect numbers up to $2\cdot10^8$.
Cohen, Graeme L., te Riele, Herman J. J.
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MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms [PDF]
Journal für die reine und angewandte Mathematik (Crelles Journal), 2010 We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of MacMahon about their ...
Andrews, George E., Rose, Simon CF
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On a modification of Set(n) [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 A modification of the set Set(n) for a fixed natural number n is introduced in the form: Set(n, f), where f is an arithmetic function. The sets Set(n,φ), Set(n,ψ), Set(n,σ) are discussed, where φ, ψ and σ are Euler's function, Dedekind's function and the
Krassimir T. Atanassov, József Sándor
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The Dirichlet divisor problem over square-free integers and unitary convolutions [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 We obtain an asymptotic formula for the sum ~D₂ of the divisors of all square-free integers less than or equal to x, with error term O(x^{1/2 + ε}). This improves the error term O(x^{3/4 + ε}) presented in [7] obtained via analytical methods.
André Pierro de Camargo
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INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH, 2023 We show that the recurrence relation deduced by Robbins and Osler et al for the sum of divisors function can be solved in terms of the complete Bell polynomials. Besides, the connection between and the number of representations of n as the sum of four triangular numbers allows obtain arecurrence relation where only participate the values of with m
R. Sivaraman +2 more
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Density bounds for the sum of divisors function [PDF]
Mathematics of Computation, 1972 Upper and lower bounds are presented for the density of the integers n n for which σ ( n ) / n ≧ x \sigma (n)/n \geqq x , where σ ( n ) \sigma (n) is the sum of divisors of n n , and
Wall, Charles R. +2 more
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ON A SUM INVOLVING CERTAIN ARITHMETIC FUNCTIONS ON PIATETSKI–SHAPIRO AND BEATTY SEQUENCES
Проблемы анализа, 2022 Let 𝑐, 𝛼, 𝛽 ∈ R be such that ...
T. Srichan
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Inequalities between some arithmetic functions, II [PDF]
Notes on Number Theory and Discrete MathematicsAs a continuation of Part I (see [1]), we offer new inequalities for classical arithmetic functions such as the Euler's totient function, the Dedekind's psi function, the sum of the positive divisors function, the number of divisors function, extended ...
Krassimir Atanassov +2 more
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Certain combinatoric Bernoulli polynomials and convolution sums of divisor functions [PDF]
Advances in Difference Equations, 2013 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Kim, Daeyeoul +1 more
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