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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 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 ...
Plot of the number of divisors of integers from 1 to 1000. Prime numbers have exactly 2 divisors, and highly composite numbers are in bold. 7 is a divisor of 42 because 7 × 6 = 42 , {\displaystyle 7\times 6=42,} so we can say 7 ∣ 42. {\displaystyle 7\mid 42.}
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.
For example, 20 has prime signature {2,1} and so the number of divisors is (2+1) × (1+1) = 6. Indeed, there are six divisors: 1, 2, 4, 5, 10 and 20. The smallest number of each prime signature is a product of primorials .
The sequence was first introduced by Paul Erdős in 1974, [1] then studied by Christopher Hooley in 1979. [2]In 2023, Dimitris Koukoulopoulos and Terence Tao proved that the sum of the first terms, = ( ) /, for . [3]
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. The integer 12 is the first abundant number.