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.

Definition

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.

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 January 2016 the largest known sphenic number is

(2^74,207,281 − 1) × (2^57,885,161 − 1) × (2^43,112,609 − 1).

It is the product of the three largest known primes.

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.

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).

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).

References

↑ 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..

Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual

Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas

With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird

Aliquot sequence-related	Untouchable Amicable Sociable Betrothed

Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
numbers

Recreational
mathematics

Base-dependent
numbers

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