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Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are in bold and superior highly composite numbers are starred. ... 10, 12, 15, 18 ...
the sequence of exponents must be non-increasing, that is ; otherwise, by exchanging two exponents we would again get a smaller number than n with the same number of divisors (for instance 18 = 2 1 × 3 2 may be replaced with 12 = 2 2 × 3 1; both have six divisors).
The divisors of n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them. Divisors and properties related to divisors are shown in table of divisors.
The numbers that these two lists have in common are the common divisors of 54 and 24, that is, ,,, Of these, the greatest is 6, so it is the greatest common divisor: (,) = Computing all divisors of the two numbers in this way is usually not efficient, especially for large numbers that have many divisors.
The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither ...
A number is divisible by a given divisor if it is divisible by the highest power of each of its prime factors. For example, to determine divisibility by 36, check divisibility by 4 and by 9. [6] Note that checking 3 and 12, or 2 and 18, would not be sufficient. A table of prime factors may be useful.
A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite.The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime.
In mathematics, Hooley's delta function (()), also called Erdős--Hooley delta-function, defines the maximum number of divisors of in [,] for all , where is the Euler's number. The first few terms of this sequence are