Sphenic number

From Free net encyclopedia

A sphenic number (Old Greek sphen = wedge) is a positive integer that is the product of three distinct prime factors. The Möbius function returns Template:Num/neg when passed any sphenic number.

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 <math>n = p \cdot q \cdot r</math>, where p, q, and r are distinct primes, then the set of divisors of n will be:

<math>\left\{ 1, \ p, \ q, \ r, \ pq, \ pr, \ qr, \ n \right\}</math>

The first few sphenic numbers are: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, ...

Currently, the largest known sphenic number is (2^30,402,457 − 1)(2^25,964,951 − 1)(2^24,036,583 − 1), i.e., the product of the three largest known Mersenne primes.

[edit]