We obtain an asymptotic formula for the number of square-free values among p 1; for primes ppx; and we apply it to derive the following asymptotic formula for LðxÞ; the number of square-free values of the Carmichael function lðnÞ for 1pnpx; LðxÞ 1⁄4 ðkþ oð1ÞÞ x ln a x ; where a 1⁄4 0:37395y is the Artin constant, and k 1⁄4 0:80328y is another absolute constant. r 2003 Elsevier Inc. All rights r...

The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Carmichael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several...

Building upon the work of Carl Pomerance and others, the central purpose of this discourse is to discuss the distribution of base-2 pseudoprimes, as well as improve upon Pomerance's conjecture regarding the Carmichael number counting function [8]. All conjectured formulas apply to any base b ≥ 2 for x ≥ x0(b). A table of base-2 pseudoprime, 2-strong pseudoprime, and Carmichael number counts up ...

a characteriation of continuity of the $p$-$lambda$-variation function is given and the helly's selection principle for $lambda bv^{(p)}$ functions is established. a characterization of the inclusion of waterman-shiba classes into classes of functions with given integral modulus of continuity is given. a useful estimate on modulus of variation of functions of class $lambda bv^{(p)}$ is found.

In this paper, we show that the counting function of the set of values of the Carmichael λ-function is x/(log x), where η = 1 − (1 + log log 2)/(log 2) = 0.08607 . . ..

In this note, we study those positive integers n which are divisible by ∑ d|n λ(d), where λ(·) is the Carmichael function.

We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZ-algebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indic...

We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indica...