Results 101 to 110 of about 41,624 (236)
, 2017 The class of Lambert series generating functions (LGFs) denoted by $L_α(q)$ formally enumerate the generalized sum-of-divisors functions, $σ_α(n) = \sum_{d|n} d^α$, for all integers $n \geq 1$ and fixed real-valued parameters $α\geq 0$. We prove new formulas expanding the higher-order derivatives of these LGFs.
openaire +3 more sources
Zaremba’s conjecture and sums of the divisor function [PDF]
Mathematics of Computation, 1993 Zaremba conjectured that given any integer m > 1 m > 1 , there exists an integer a > m a > m with a relatively prime to m such that the simple continued fraction [ 0 , c 1 ,
openaire +2 more sources
Some results concerning exponential divisors
International Journal of Mathematics and Mathematical Sciences, 1988 If the natural number n has the canonical form p1a1p2a2…prar then d=p1b1p2b2…prbr is said to be an exponential divisor of n if bi|ai for i=1,2,…,r. The sum of the exponential divisors of n is denoted by σ(e)(n). n is said to be an e-perfect number if σ(e)
Peter Hagis
doaj +1 more source
Degree theory for 4‐dimensional asymptotically conical gradient expanding solitons
Communications on Pure and Applied Mathematics, Volume 79, Issue 5, Page 1151-1298, May 2026.Abstract We develop a new degree theory for 4‐dimensional, asymptotically conical gradient expanding solitons. Our theory implies the existence of gradient expanding solitons that are asymptotic to any given cone over S3$S^3$ with non‐negative scalar curvature. We also obtain a similar existence result for cones whose link is diffeomorphic to S3/Γ$S^3/\
Richard H. Bamler, Eric Chen
wiley +1 more source
Probabilistic Methods on Erdos Problems [PDF]
, 2013 The study of perfect numbers dates back to Euler and Mersenne. A perfect number is a number that is equal to the sum of its proper divisors which are said to include the multiplicative unit 1.
Gilbert, Jesse
core +1 more source
Time‐resolved XPCS analysis across broad time‐scales using multi‐tau two‐time correlations
Journal of Synchrotron Radiation, Volume 33, Issue 3, Page 698-707, May 2026.A scalable multi‐tau two‐time correlation framework enables efficient, time‐resolved XPCS analysis of long duration high‐frame‐rate modern synchrotron data while preserving sensitivity to non‐stationary dynamics.We present a multi‐tau two‐time correlation (MT‐2TC) analysis for X‐ray photon correlation spectroscopy that enables efficient analysis of ...
Fabio Brugnara +13 more
wiley +1 more source
Divisibility and sequence properties of σ⁺ and φ⁺ [PDF]
Notes on Number Theory and Discrete MathematicsInspired by Lehmer’s and Deaconescu’s conjectures, as well as various analogue problems concerning Euler’s totient function φ(n), Schemmel’s totient function S₂(n), Jordan totient function Jₖ, and the unitary totient function φ*(n), we investigate ...
Sagar Mandal
doaj +1 more source
On the ratio of the sum of divisors and Euler’s Totient Function I [PDF]
, 2013 We prove that the only solutions to the equation σ(n)=2φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n)=2φ(n) with Ω(n)≤ k, and there are at most 22k+k-k squarefree ...
Broughan, Kevin A., Delbourgo, Daniel
core +2 more sources
Acta Crystallographica Section A, Volume 82, Issue 3, Page 145-162, May 2026.General formulas are presented that allow for the enumeration of polytypes based on translationally equivalent layers and two equivalent arrangements of adjacent layers involving distinct possible stacking vectors, t1 and t2. The results have been applied to the polytypism among two different polysomes of the family of so‐called silico‐ferrites of ...
Michael Francesco Salzmann +3 more
wiley +1 more source
The sum of the unitary divisor function [PDF]
, 2014 This article establishes a new upper bound on the function $\sigma^{*}(n)$, the sum of all coprime divisors of $n$. The article concludes with two questions concerning this function.Comment: 6 pages, to appear in Publ. Inst. Math. (Beograd) (N.S.
Trudgian, Tim
core