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65536 is the natural number following 65535 and preceding 65537.. 65536 is a power of two: (2 to the 16th power).. 65536 is the smallest number with exactly 17 divisors (but there are smaller numbers with more than 17 divisors; e.g., 180 has 18 divisors) (sequence A005179 in the OEIS).
The only known powers of 2 with all digits even are 2 1 = 2, 2 2 = 4, 2 3 = 8, 2 6 = 64 and 2 11 = 2048. [12] The first 3 powers of 2 with all but last digit odd is 2 4 = 16, 2 5 = 32 and 2 9 = 512. The next such power of 2 of form 2 n should have n of at least 6 digits.
If n is an even superperfect number, then n must be a power of 2, 2 k, such that 2 k+1 − 1 is a Mersenne prime. [1] [2] It is not known whether there are any odd superperfect numbers. An odd superperfect number n would have to be a square number such that either n or σ(n) is divisible by at least three distinct primes. [2]
When expressed as exponents, the geometric series is: 2 0 + 2 1 + 2 2 + 2 3 + ... and so forth, up to 2 63. The base of each exponentiation, "2", expresses the doubling at each square, while the exponents represent the position of each square (0 for the first square, 1 for the second, and so on.). The number of grains is the 64th Mersenne number.
Here, 243 is the 5th power of 3, ... Also unlike addition and multiplication, exponentiation is not associative: for example, (2 3) 2 = 8 2 = 64, whereas 2 (3 2) ...
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: 7 = 7 1 , 9 = 3 2 and 64 = 2 6 are prime powers, while 6 = 2 × 3 , 12 = 2 2 × 3 and 36 = 6 2 = 2 2 × 3 2 are not.
For example, to find the seventh Fortunate number, one would first calculate the product of the first seven primes (2, 3, 5, 7, 11, 13 and 17), which is 510510. Adding 2 to that gives another even number, while adding 3 would give another multiple of 3. One would similarly rule out the integers up to 18.
The aliquot sum of a power of two (2 n) is always one less than the power of two itself, therefore the aliquot sum of 64 is 63, within an aliquot sequence of two composite members (64, 63, 41, 1, 0) that are rooted in the aliquot tree of the thirteenth prime, 41. [2] 64 is: the smallest number with exactly seven divisors, [3]
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