Sphenic number

In number theory, a sphenic number (from Ancient Greek: σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers.

A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. This definition is more stringent than simply requiring the integer to have exactly three prime factors. For instance, 60 = 2² × 3 × 5 has exactly 3 prime factors, but is not sphenic.

The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are

As of January 2016^[ref] the largest known sphenic number is

It is the product of the three largest known primes.

All sphenic numbers have exactly eight divisors. If we express the sphenic number as ${\displaystyle n=p\cdot q\cdot r}$, where p, q, and r are distinct primes, then the set of divisors of n will be:

The converse does not hold. For example, 24 is not a sphenic number, but it has exactly eight divisors.

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

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