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 $n=p\cdot q\cdot r$, 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 cyclotomic polynomials $\Phi _{n}(x)$, taken over all sphenic numbers $n$, may contain arbitrarily large coefficients (for $n$ a product of two primes the coefficients are $\pm 1$ or 0).

The first few sphenic numbers are: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... (sequence A007304 in the 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 consecutive positive integer is divisible by 4 = 2×2 and therefore not squarefree.

As of June 2009 the largest known sphenic number is (2^43,112,609 − 1) × (2^42,643,801 − 1) × (2^37,156,667 − 1), i.e., the product of the three largest known primes.

See also

• Semiprimes, products of two prime numbers.

External links

• Sphenic numbers from On-Line Encyclopedia of Integer Sequences.