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The number of perfect numbers less than n is less than , where c > 0 is a constant. [54] In fact it is (), using little-o notation. [55] Every even perfect number ends in 6 or 28, base ten; and, with the only exception of 6, ends in 1 in base 9.
Perfect numbers are natural numbers that equal the sum of their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4] Euclid proved c. 300 BCE that every prime expressed as M p = 2 p − 1 has a corresponding perfect number ...
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.
Notably, absent consensus, please do not add articles about individual perfect numbers themselves (such as 6). Pages in category "Perfect numbers" The following 11 pages are in this category, out of 11 total.
A minimal perfect hash function is a perfect hash function that maps n keys to n consecutive integers – usually the numbers from 0 to n − 1 or from 1 to n. A more formal way of expressing this is: Let j and k be elements of some finite set S .
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 ...
Every perfect number is also -perfect. [1] However, there are numbers such as 24 which are -perfect but not perfect. The only known -perfect number with three distinct prime factors is 126 = 2 · 3 2 · 7. [2] Every number of form 2^(n - 1) * (2^n - 1) * (2^n)^m where m >= 0, where 2^n - 1 is Prime, are Granville Numbers.
Hence the rate of Hamming codes is R = k / n = 1 − r / (2 r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2 r − 1.