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

Note that this definition is more stringent than simply requiring the integer to have exactly three prime factors; e.g. 60 = 2² × 3 × 5 has exactly 3 prime factors, but is not sphenic.

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:

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

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).

The first few sphenic numbers are: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ... (sequence A007304 in OEIS)

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), i.e., the product of the three largest known primes.

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 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..