A sphenic number (Old Greek sphen = wedge) is a positive integer that is the product of three distinct prime factors. The Möbius function returns 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 $n\; =\; p\; \backslash cdot\; q\; \backslash cdot\; r$, where p, q, and r are distinct primes, then the set of divisors of n will be:

$\backslash left\backslash \{\; 1,\; \backslash \; p,\; \backslash \; q,\; \backslash \; r,\; \backslash \; pq,\; \backslash \; pr,\; \backslash \; qr,\; \backslash \; n\; \backslash right\backslash \}$

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.

External links

• Sphenic numbers from On-Line Encyclopedia of Integer Sequences.

Integer sequences | Prime numbers

Nombre sphénique | Numero sfenico | מספר ספני | Szfenikus szám | Sphenisch getal | Número esfênico | Klinasto število