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In [[number theory]], a '''sphenic number''' (from {{lang-grc|σφήνα}}, 'wedge') is a [[natural number|positive integer]] that is the product of three distinct [[prime number]]s.


Note that this definition is more stringent than simply requiring the integer to have exactly three [[prime factor]]s; e.g. 60 = 2<sup>2</sup> × 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 <math>n = p \cdot q \cdot r</math>, where ''p'', ''q'', and ''r'' are distinct primes, then the set of divisors of ''n'' will be:
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:<math>\left\{ 1, \ p, \ q, \ r, \ pq, \ pr, \ qr, \ n \right\}.</math>
 
All sphenic numbers are by definition [[Square-free integer|squarefree]], because the prime factors must be distinct.
 
The [[Möbius function]] of any sphenic number is &minus;1.
 
The [[cyclotomic polynomials]] <math>\Phi_n(x)</math>, taken over all sphenic numbers ''n'', may contain arbitrarily large coefficients<ref>Emma Lehmer, "On the magnitude of the coefficients of the cyclotomic polynomial", ''Bulletin of the American Mathematical Society'' 42 (1936), no. 6, pp. 389–392.[http://www.projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183498920].</ref> (for ''n'' a product of two primes the coefficients are <math>\pm 1</math> or 0).
 
The first few sphenic numbers are: [[30 (number)|30]], [[42 (number)|42]], [[66 (number)|66]], [[70 (number)|70]], [[78 (number)|78]], [[102 (number)|102]], [[105 (number)|105]], [[110 (number)|110]], [[114 (number)|114]], [[130 (number)|130]], [[138 (number)|138]], [[154 (number)|154]], [[165 (number)|165]], ... {{OEIS|id=A007304}}
 
{{As of|2013|2|url=http://primes.utm.edu/top20/page.php?id=3}} the largest known sphenic number is (2<sup>57,885,161</sup> − 1) × (2<sup>43,112,609</sup> − 1) × (2<sup>42,643,801</sup> − 1), i.e., the product of the three [[largest known prime]]s.
 
==Consecutive sphenic numbers==
The first case of two consecutive sphenic integers 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 every fourth consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree.
 
The numbers 2013, 2014, 2015 are all sphenic. The next three consecutive sphenic years will be 2665, 2666 and 2667 {{OEIS|id=A165936}}.
 
== See also ==
* [[Semiprime]]s, products of two [[prime number]]s.
* [[Almost prime]]
 
== References ==
{{reflist}}
 
 
{{Divisor classes}}
{{Classes of natural numbers}}
[[Category:Integer sequences]]
[[Category:Prime numbers]]
 
 
{{numtheory-stub}}

Latest revision as of 12:28, 11 January 2015


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