A highly composite number is a number (positive integer) with more divisors than numbers less than it. The first few highly composite numbers with prime factorizations (bases omitted) and number of divisors are
1 - 1 2 1 2 4 2 3 6 1 1 4 12 2 1 6 24 3 1 8 36 2 2 9 48 4 1 10 60 2 1 1 12 120 3 1 1 16 180 2 2 1 18 240 4 1 1 20 360 3 2 1 24 720 4 2 1 30 840 3 1 1 1 32 1260 2 2 1 1 36 1680 4 1 1 1 40 2520 3 2 1 1 48
A special highly composite number is a highly composite number that is a factor of all larger highly composite numbers. Prove the following theorem: There are exactly six special highly composite numbers, 1, 2, 6, 12, 60, and 2520. A proof is in the December 1991 issue of Mathematics Magazine. Find a highly composite number (other than 1) that is not the product of a prime number and another highly composite number. The largest highly composite number I found could be represented 16 9 6 5, 2 5 19 570. This is an abbreviated form of the prime factorization with bases omitted. The 2, 5, 19, and 570 represent 2, 5, 19, and 570 exponents of 4, 3, 2, and 1, respectively.