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.
The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are
As of October 2020[ref] the largest known sphenic number is
- (282,589,933 − 1) × (277,232,917 − 1) × (274,207,281 − 1).
It is the product of the three largest known primes.
All sphenic numbers have exactly eight divisors. If we express the sphenic number as , where p, q, and r are distinct primes, then the set of divisors of n will be:
The converse does not hold. For example, 24 is not a sphenic number, but it has exactly eight divisors.
All sphenic numbers are by definition squarefree, because the prime factors must be distinct.
The Möbius function of any sphenic number is −1.
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).
- 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..