Dozenal Wiki
Advertisement

A sphenic number is the product of three distinct prime numbers. The word "sphenic" means "wedge-shaped".

The first sphenic numbers are 26, 36, 56, 5X, 66, 86, 89, 92, 96, XX, E6, 10X, 119, 122, 126, 132, 136, 13X, 143, 166, 172, 173, 17X, 186, 193, 196, 1X2, 1X9, 1E6, 1E9, 1EX, 202, 21X, 226, 22X, 249, 256, 259, 266, 26X, 272, 281, 293, 2296, 29X, 2X2, 2XX, 2E6, 2E9, 2EX.

All sphenic numbers are square-free, since by definition they are the product of three distinct primes and numbers which are products of distinct primes are square-free.

All sphenic numbers have exactly 8 divisors. The divisors of any sphenic number n factorized as p × q × r have the divisors expressed as {1, p, q, r, pq, pr, qr, n}. The converse does not hold. For example, 20 = 23 × 3 has 8 divisors, but it is not a sphenic number.

The first case of two consecutive sphenic numbers is 172 = 2 × 5 × 1E and 173 = 3 × 7 × E. The first case of three consecutive sphenic numbers is 911 = 7 × E × 15, 912 = 2 × 5 × XE and 913 = 3 × 17 × 1E. There is no case for more than three, because every fourth integer is divisible by 22 = 4 and therefore is not square-free. (or the property of sphenic numbers in dozenal: sphenic numbers end with 1, 2, 3, 5, 6, 7, 9, X, E in dozenal, thus there cannot be 4 consecutive sphenic number)