semiprime
Semiprime
2002-06-28 11:44:52
2006-07-09 17:01:19
Definition
almost prime
number theory
primes
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A composite number which is the product of two (possibly equal) primes is called \emph{semiprime}. Such numbers are sometimes also called 2-\emph{almost primes}. For example:
\begin{itemize}
\item 1 is not a semiprime because it is not a composite number or a prime,
\item 2 is not a semiprime, as it is a prime,
\item 4 is a semiprime, since $4 = 2\cdot 2$,
\item 8 is not a semiprime, since it is a product of three primes ($8 = 2\cdot 2\cdot 2$),
\item 2003 is not a semiprime, as it is a prime,
\item 2005 is a semiprime, since $2005 = 5\cdot 401$,
\item 2007 is not a semiprime, since it is a product of three primes ($2007 = 3\cdot 3\cdot 223$).
\end{itemize}
The first few semiprimes are $4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, \ldots$ (\PMlinkexternal{Sloane's sequence A001358}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001358}
). The Moebius function $\mu(n)$ for semiprimes can be only equal to 0 or 1. If we form an integer sequence of values of $\mu(n)$ for semiprimes we get a binary sequence: $0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, \ldots$. (\PMlinkexternal{Sloane's sequence A072165}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=072165}
).
All the squares of primes are also semiprimes. The first few squares of primes are then $4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, \ldots$. (\PMlinkexternal{Sloane's sequence A001248}{http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001248}
). The Moebius function $\mu(n)$ for the squares of primes is always equal to 0 as it is equal to 0 for all squares.