# 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.

## Definition

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 = 2^{2} × 3 × 5 has exactly 3 prime factors, but is not sphenic.

## Examples

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 the largest known sphenic number is

- (2
^{74,207,281 }− 1) × (2^{57,885,161}− 1) × (2^{43,112,609}− 1).

It is the product of the three largest known primes.

## Divisors

All sphenic numbers have exactly eight divisors. If we express the sphenic number as , 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.

## Properties

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 , taken over all sphenic numbers *n*, may contain arbitrarily large coefficients^{[1]} (for *n* a product of two primes the coefficients are or 0).

## Consecutive sphenic numbers

The first case of two consecutive sphenic integers 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.

The numbers 2013 (3×11×61), 2014 (2×19×53), and 2015 (5×13×31) are all sphenic. The next three consecutive sphenic years will be 2665 (5×13×41), 2666 (2×31×43) and 2667 (3×7×127) (sequence A165936 in the OEIS).

## See also

- Semiprimes, products of two prime numbers.
- Almost prime

## References

- ↑ Emma Lehmer, "On the magnitude of the coefficients of the cyclotomic polynomial",
*Bulletin of the American Mathematical Society*42 (1936), no. 6, pp. 389–392..