
In mathematics, a sphenic number (Old Greek sphen = wedge) is a positive integer which is the product of three distinct prime numbers. Note that this definition is more stringent than simply requiring the integer to have exactly three prime factors; e.g. 60 = 22 × 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*q*r, where p, q, and r are distinct primes, then the set of divisors of n will be: { 1, p, q, r, pq, pr, qr, n }. 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, ... (sequence A007304 in OEIS) 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 currently (2^{32,582,657} − 1) × (2^{30,402,457} − 1) × (2^{25,964,951} − 1), i.e., the product of the three largest known primes. Links * Sphenic numbers from OnLine Encyclopedia of Integer Sequences. Retrieved from "http://en.wikipedia.org/" 
