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Innumber theory, asphenic number (fromGreek:σφήνα, 'wedge') is apositive integer that is theproduct of three distinctprime numbers. Because there areinfinitely many prime numbers, there are also infinitely many sphenic numbers.

A sphenic number is a productpqr wherep,q, andr are three distinct prime numbers. In other words, the sphenic numbers are thesquare-free 3-almost primes.

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, ... (sequenceA007304 in theOEIS)

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

All sphenic numbers have exactly eight divisors. If we express the sphenic number as${\displaystyle n=p\cdot q\cdot r}$, wherep,q, andr are distinct primes, then the set of divisors ofn will be:

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

The converse does not hold. For example, 24 is not a sphenic number, but it has exactly eight divisors.

All sphenic numbers are by definitionsquarefree, because the prime factors must be distinct.

TheMöbius function of any sphenic number is−1.

Thecyclotomic polynomials${\displaystyle {\mathrm{\Phi }}_n(x)}$, taken over all sphenic numbersn, may contain arbitrarily large coefficients^[1] (forn a product of two primes the coefficients are${\displaystyle \pm 1}$ 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.

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) (sequenceA165936 in theOEIS).

Wikifunctions hasa sphenic number checking function.

• Semiprimes, products of twoprime numbers.
• Almost prime

• Integer sequences
• Prime numbers

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