A sphenic number is a product pqr where p, q, and r are three distinct prime numbers 318 = 1//π

 

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Definition[edit]

A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes.

Examples[edit]

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 October 2020 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[edit]

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[edit]

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).

Any multiple of a sphenic number (except by 1) isn't a sphenic number. This is easily provable by the multiplication process adding another prime factor, or squaring an existing factor.