48 (nombre)

48 (forty-eight) is the natural number following 47 and preceding 49. It is one third of a gross, or four dozens. Forty-eight is the double factorial of 6, a highly composite number. Like all other multiples of 6, it is a semiperfect number. 48 is the second 17-gonal number. 48 is the smallest number with exactly ten divisors, and the first multiple of 12 not to be a sum of twin primes. The Sum of Odd Anti-Factors of 48 = number * (n/2) where n is an Odd number. So, 48 is an Odd Anti-Factor Hemiperfect Number. Other such numbers include 6048, 38688, 82132, 975312, etc. Odd Anti-Factors of 48 = 5, 19 Sum of Odd Anti-Factors = 5 + 19 = 24 = 48 * 1/2 There are 11 solutions to the equation φ(x) = 48, namely 65, 104, 105, 112, 130, 140, 144, 156, 168, 180 and 210. This is more than any integer below 48, making 48 a highly totient number. Since the greatest prime factor of 482 + 1 = 2305 is 461, which is clearly more than twice 48, 48 is a Størmer number. 48 is a Harshad number in base 10. It has 24, 2, 12, and 4 as factors. The atomic number of cadmium. The number of Ptolemaic constellations. The number of symmetries of a cube. Messier object M48, a magnitude 5.5 open cluster in the constellation Hydra. The New General Catalogue object NGC 48, a barred spiral galaxy in the constellation Andromeda. The prophecies of 48 Jewish prophets and 7 prophetesses were recorded in the Tanakh for posterity. According to the Mishnah, Torah wisdom is acquired via 48 ways (Pirkei Avoth 6:6). In Buddhism, Amitabha Buddha had made 48 great vows and promises to provide ultimate salvation to countless beings through countless eons, with benefits said to be available merely by thinking about his name with Nianfo practice. He is thus hailed as "King of Buddhas" through such skillful compassion and became a popular and formal refuge figure in Pureland Buddhism. Johann Sebastian Bach's Well-Tempered Clavier is informally known as because it consists of a prelude and a fugue in each major and minor key, for a total of forty-eight pieces.