| ||||
---|---|---|---|---|
Cardinal | forty-eight | |||
Ordinal | 48th (forty-eighth) | |||
Factorization | 24 × 3 | |||
Divisors | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | |||
Greek numeral | ΜΗ´ | |||
Roman numeral | XLVIII | |||
Binary | 1100002 | |||
Ternary | 12103 | |||
Senary | 1206 | |||
Octal | 608 | |||
Duodecimal | 4012 | |||
Hexadecimal | 3016 |
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 is given by the equations for even , and for odd . [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 [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:
| ||||
---|---|---|---|---|
Cardinal | forty-eight | |||
Ordinal | 48th (forty-eighth) | |||
Factorization | 24 × 3 | |||
Divisors | 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 | |||
Greek numeral | ΜΗ´ | |||
Roman numeral | XLVIII | |||
Binary | 1100002 | |||
Ternary | 12103 | |||
Senary | 1206 | |||
Octal | 608 | |||
Duodecimal | 4012 | |||
Hexadecimal | 3016 |
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 is given by the equations for even , and for odd . [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 [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: