SphenicNumber 2006-08-17 12:06:10 2006-11-15 10:18:27 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given three primes $p < q < r$, the composite integer $pqr$ is a {\em sphenic number}. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, $\ldots$ listed in A007304 of Sloane's OEIS. The divisors of a sphenic number therefore are $1, p, q, r, pq, pr, qr, pqr$. Furthermore, $\mu(pqr) = (-1)^3$ (where $\mu$ is the M\"obius function), $\tau(pqr) = 8$ (where $\tau$ is the divisor function) and $\Omega(pqr) = \omega(pqr) = 3$ (where $\Omega$ and $\omega$ are the \PMlinkname{number of (nondistinct) prime factors function}{NumberOfNondistinctPrimeFactorsFunction} and the number of distinct prime factors function, respectively). The largest known sphenic number at any time is usually the product of the three largest known Mersenne primes.