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A highly composite number is a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N. For example, 6 is highly composite because d(6)=4 and d(n)=1,2,2,3,2 for n=1,2,3,4,5 respectively.
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. In the SVG file, hover over a bar to see its statistics. The tables below list all of the divisors of the numbers 1 to 1000.
If the number of digits is even, add the first and subtract the last digit from the rest. The result must be divisible by 11. 918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 + 1 − 5 = 77 = 7 × 11. If the number of digits is odd, subtract the first and last digit from the rest. The result must be divisible by 11.
A number that has the same number of digits as the number of digits in its prime factorization, including exponents but excluding exponents equal to 1. A046758: Extravagant numbers: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, ... A number that has fewer digits than the number of digits in its prime factorization (including ...
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 ...
New prime is 16 million digits larger than previous one
The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5. [ 11 ] The set of all primes is sometimes denoted by P {\displaystyle \mathbf {P} } (a boldface capital P) [ 12 ] or by P {\displaystyle \mathbb {P} } (a blackboard bold ...
Numbers p and q like this can be computed with the extended Euclidean algorithm. gcd(a, 0) = | a |, for a ≠ 0, since any number is a divisor of 0, and the greatest divisor of a is | a |. [2] [5] This is usually used as the base case in the Euclidean algorithm. If a divides the product b⋅c, and gcd(a, b) = d, then a/d divides c.