Sphenic number

In number theory, a sphenic number (from Template:Langx, 'wedge') is a positive integer that is the product of three distinct prime numbers.Template:RTemplate:R For example, since 2, 3, and 73 are all prime, 438 is a sphenic number because Template:Math.Template:R Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers.

Definition

A sphenic number is a product Template:Mvar where Template:Mvar, Template:Mvar, and Template:Mvar are three distinct prime numbers.Template:R In other words, the sphenic numbers are the square-free 3-almost primes.

Examples

The smallest sphenic number is Template:Math, 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, ... Template:OEIS

The largest known sphenic number at any time can be obtained by multiplying together the three largest known primes.

Divisors

All sphenic numbers have exactly eight divisors.Template:R If we express the sphenic number as Template:Math, where Template:Mvar, Template:Mvar, and Template:Mvar are distinct primes, then the set of divisors of Template:Mvar will be:

$\{1,p,q,r,pq,pr,qr,n\}.$Template:R

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

Any multiple of a sphenic number (except by 1) is not sphenic. This is easily provable by the multiplication process at a minimum adding another prime factor, or raising an existing factor to a higher power.

Consecutive sphenic numbers

The first case of two consecutive sphenic integers is Template:Math and Template:Math. The first case of three is Template:Math, Template:Math, and Template:Math. There is no case of more than three, because every fourth consecutive positive integer is divisible by Template:Math and therefore not squarefree Template:OEIS.

The numbers Template:Math, Template:Math, and Template:Math are all sphenic. The next three consecutive sphenic years will be Template:Math, Template:Math and Template:Math Template:OEIS.

See also

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

References

Template:Divisor classes Template:Classes of natural numbers