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,[1][2] a highly composite number.[3] Like all other multiples of 6, it is a semiperfect number.[4] 48 is the smallest non-trivial 17-gonal number.[5]
48 is the smallest number with exactly ten divisors,[6] and the first multiple of 12 not to be a sum of twin primes.
There are eleven solutions to the equation φ(x) = 48, namely {65, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210}. This is more than any integer below it, making 48 a highly totient number.[7] On the other hand, the totient of 48 is 16,[8] a third of its numeric value, that is also the number of divisors of 168,[9] the seventeenth record for sum-of-divisors of natural numbers where 48 specifically sets the sixteenth such record value, of 124.[10]
Since the greatest prime factor of 482 + 1 = 2305 is 461, which is clearly more than twice 48, 48 is a Størmer number.[11]
48 is a Harshad number in decimal,[12] as it is divisible by 4+8 = 12.
By a classical result of Honsberger, the number of incongruent integer-sided triangles of perimeter m {\displaystyle m} is given by the equations m 2 48 {\displaystyle {\tfrac {m^{2}}{48}}} for even m {\displaystyle m} , and ( m + 3 ) 2 48 {\displaystyle {\tfrac {(m+3)^{2}}{48}}} for odd m {\displaystyle m} .[13]
48 is the order of full octahedral symmetry, which describes three-dimensional mirror symmetries associated with the regular octahedron and cube. 48 is also twice the order of full tetrahedral symmetry (24).
48 is the floor and nearest-integer value of the ninth imaginary part of non-trivial zeroes in the Riemann zeta function (see, Riemann hypothesis).[14][15] Among the nine first such floor and ceiling values, this is the closest to an integer, differing from 48 by a value of around + 0.0051508811671597279 … … --> {\displaystyle +{\text{ }}0.0051508811671597279\dots } [16]
Meanwhile, the fifth such ceiling value is 33,[17] which is the smallest of only three numbers to hold a sum-of-divisors of 48 (the others are 35 and 47).[18] The composite index of 48 represents the fifth floor value in this sequence, 32.[19][14] The smallest floor and ceiling values in the Riemann zeta function are 14 and 15, which are the two smallest numbers (of three total) to hold a sum-of-divisors of 24 (half 48).
Forty-eight may also refer to:
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