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 , where p, q, and r are distinct primes, then the set of divisors of n will be:

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 , taken over all sphenic numbers n, may contain arbitrarily large coefficients^[1] (for n a product of two primes the coefficients are 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)

The only abundant sphenic number that is not divisible by 6 is 70.^{[citation needed]}

As of February 2013^[ref] the largest known sphenic number is (2^57,885,161 − 1) × (2^43,112,609 − 1) × (2^42,643,801 − 1), i.e., the product of the three largest known primes.

Consecutive sphenic numbers[edit]

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[edit]

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

References[edit]

Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number Factorization forms Prime number Composite number Semiprime number Pronic number Sphenic number Square-free number Powerful number Perfect power Achilles number Smooth number Regular number Rough number Unusual number Constrained divisor sums Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hemiperfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Practical number Erdős–Nicolas number With many divisors Abundant number Primitive abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Weird number Aliquot sequence-related Untouchable number Amicable number Sociable number Betrothed number Other sets Deficient number Friendly number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number