Positive integer that is the product of three distinct prime numbers

In number theory, a **sphenic number** (from Ancient Greek: σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers.

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Definition

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.

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Examples

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^{[ref]} 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.

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Divisors

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:

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

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

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

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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).