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1. CMB 2009 (vol 52 pp. 3)
Carmichael Numbers with a Square Totient Let $\varphi$ denote the Euler function. In this paper, we show that
for all large $x$ there are more than $x^{0.33}$ Carmichael numbers
$n\le x$ with the property that $\varphi(n)$ is a perfect square. We
also obtain similar results for higher powers.
Categories:11N25, 11A25 |
2. CMB 2008 (vol 51 pp. 3)
3. CMB 2005 (vol 48 pp. 211)
The Distribution of Totatives The integers coprime to $n$ are called the {\it totatives} \rm of $n$.
D. H. Lehmer and Paul Erd\H{o}s were interested in understanding when
the number of totatives between $in/k$ and $(i+1)n/k$ are $1/k$th of
the total number of totatives up to $n$. They provided criteria in
various cases. Here we give an ``if and only if'' criterion which
allows us to recover most of the previous results in this literature
and to go beyond, as well to reformulate the problem in terms of
combinatorial group theory. Our criterion is that the above holds if
and only if for every odd character $\chi \pmod \kappa$ (where
$\kappa:=k/\gcd(k,n/\prod_{p|n} p)$) there exists a prime $p=p_\chi$
dividing $n$ for which $\chi(p)=1$.
Categories:11A05, 11A07, 11A25, 20C99 |
4. CMB 2002 (vol 45 pp. 109)
The Distribution of Totatives D.~H.~Lehmer initiated the study of the distribution of totatives, which
are numbers coprime with a given integer. This led to various problems
considered by P.~Erd\H os, who made a conjecture on such distributions.
We prove his conjecture by establishing a theorem on the ordering of
residues.
Keywords:Euler's function, totatives Categories:11A05, 11A07, 11A25 |
5. CMB 1997 (vol 40 pp. 498)
Matrix transformations based on Dirichlet convolution This paper is a study of summability methods that are based
on Dirichlet convolution. If $f(n)$ is a function on positive integers
and $x$ is a sequence such that $\lim_{n\to \infty} \sum_{k\le n}
{1\over k}(f\ast x)(k) =L$, then $x$ is said to be {\it $A_f$-summable\/}
to $L$. The necessary and sufficient condition for the matrix $A_f$ to
preserve bounded variation of sequences is established. Also, the
matrix $A_f$ is investigated as $\ell - \ell$ and $G-G$ mappings. The
strength of the $A_f$-matrix is also discussed.
Categories:11A25, 40A05, 40C05, 40D05 |