Sphenic number

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 On-Line Encyclopedia of Integer Sequences.