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In mathematics, a sphenic number (Old Greek sphen = wedge) is a positive integer which is the product of three distinct prime numbers.## External links

Note that this definition is more stringent than simply requiring the integer to have exactly three prime factors; e.g. 60 = 2^{2} × 3 × 5 has exactly 3 prime factors, but is not sphenic.

All sphenic numbers have exactly eight divisors. If we express the sphenic number as $n\; =\; p\; cdot\; q\; cdot\; r$, where p, q, and r are distinct primes, then the set of divisors of n will be:

- $left\{\; 1,\; p,\; q,\; r,\; pq,\; pr,\; qr,\; n\; right\}.$

All sphenic numbers are by definition squarefree, because the prime factors must be distinct.

The Möbius function of any sphenic number is −1.

The first few sphenic numbers are: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ...

The first case of two consecutive integers which are sphenic numbers is 230 = 2×5×23 and 231 = 3×7×11. The first case of three is 1309 = 7×11×17, 1310 = 2×5×131, and 1311 = 3×19×23. There is no case of more than three, because one of every four consecutive integers is divisible by 4 = 2×2 and therefore not squarefree.
the largest known sphenic number is (2^{43,112,609} − 1) × (2^{37,156,667} − 1) × (2^{32,582,657} − 1), i.e., the product of the three largest known primes.

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Last updated on Sunday September 28, 2008 at 05:44:43 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday September 28, 2008 at 05:44:43 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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