Seventy-two is a pronic number, as it is the product of 8 and 9. It is the smallest Achilles number, as it's a powerful number that is not itself a power. 72 is an abundant number. With exactly twelve positive divisors, including 12 (one of only two sublime numbers), 72 is also the twelfth member in the sequence of refactorable numbers. As no smaller number has more than 12 divisors, 72 is a largely composite number.72 has an Euler totient of 24. It is a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer under 72. It is equal to the sum of its preceding smaller highly totient numbers 24 and 48, and contains the first six highly totient numbers 1, 2, 4, 8, 12 and 24 as a subset of its proper divisors. 144, or twice 72, is also highly totient, as is 576, the square of 24. While 17 different integers have a totient value of 72, the sum of Euler's totient function φ(x) over the first 15 integers is 72. It also is a perfect indexed Harshad number in decimal (twenty-eighth), as it is divisible by the sum of its digits (9).
