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 = 2² × 3 × 5 has exactly 3 prime factors, but is not sphenic.

Examples

The sphenic numbers are the square-free 3-almost primes.

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

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... (sequence A007304 in the OEIS)

As of January 2019 the largest known sphenic number is

(2^82,589,933 − 1) × (2^77,232,917 − 1) × (2^74,207,281 − 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 ${\displaystyle n=p\cdot q\cdot r}$, where p, q, and r are distinct primes, then the set of divisors of n will be:

${\displaystyle \left\{1,\ p,\ q,\ r,\ pq,\ pr,\ qr,\ n\right\}.}$

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

1. 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..