Concept
72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or 6 dozen (i.e., 60 in duodecimal). 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. 72 has a Euler totient of 24, which makes it a highly totient number, as there are 17 solutions to the equation φ(x) = 72, more than any integer below 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). 72 is the second multiple of 12, after 48, that is not a sum of twin primes. It is, however, the sum of four consecutive primes (13 + 17 + 19 + 23), as well as the sum of six consecutive primes (5 + 7 + 11 + 13 + 17 + 19). 72 is the smallest number whose fifth power is the sum of five smaller fifth powers: 195 + 435 + 465 + 475 + 675 = 725. 72 is the number of distinct {7/2} magic heptagrams, all with a magic constant of 30. 72 is the sum of the eighth row of Lozanić's triangle. 72 is the number of degrees in the central angle of a regular pentagon, which is constructible with a compass and straight-edge. 72 plays a role in the Rule of 72 in economics when approximating annual compounding of interest rates of a round 6% to 10%, due in part to its high number of divisors. Inside Lie algebras: 72 is the number of vertices of the six-dimensional 122 polytope, which also contains as facets 720 edges, 702 polychoral 4-faces, of which 270 are four-dimensional 16-cells, and two sets of 27 demipenteract 5-faces.