Sphenic number

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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.

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 $n = p \cdot q \cdot r$ , 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. For this reason, neither 8 nor 12 nor 18 is sphenic.

The Möbius function returns −1 when passed any sphenic number.

The first few sphenic numbers are: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, ... (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 every fourth integer is divisible by 4 = 2×2 and therefore not squarefree.

As of 2006, the largest known sphenic number is (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. At this time all three are Mersenne primes.