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In number theory, a narcissistic number [1] [2] (also known as a pluperfect digital invariant (PPDI), [3] an Armstrong number [4] (after Michael F. Armstrong) [5] or a plus perfect number) [6] in a given number base is a number that is the sum of its own digits each raised to the power of the number of digits.
Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has Ω( n ) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS ).
This is a list of articles about prime numbers.A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers.
6174 is a 7-smooth number, i.e. none of its prime factors are greater than 7. 6174 can be written as the sum of the first three powers of 18: 18 3 + 18 2 + 18 1 = 5832 + 324 + 18 = 6174, and coincidentally, 6 + 1 + 7 + 4 = 18. The sum of squares of the prime factors of 6174 is a square: 2 2 + 3 2 + 3 2 + 7 2 + 7 2 + 7 2 = 4 + 9 + 9 + 49 + 49 ...
A 176-digit cofactor of 11 281 + 1 was factored between February and May 2005 using machines at NTT and Rikkyo University in Japan. [1] The 663-bit (200-digit) RSA-200 challenge number was factored between December 2003 and May 2005, using a cluster of 80 Opteron processors at BSI in Germany; the announcement was made on 9 May 2005. [2]
Two types of factors can be derived from a Cunningham number without having to use a factorization algorithm: algebraic factors of binomial numbers (e.g. difference of two squares and sum of two cubes), which depend on the exponent, and aurifeuillean factors, which depend on both the base and the exponent.
In number theory, the prime omega functions and () count the number of prime factors of a natural number . Thereby (little omega) counts each distinct prime factor, whereas the related function () (big omega) counts the total number of prime factors of , honoring their multiplicity (see arithmetic function).
Prime factors of one above the base: 11 Other prime factors: 7 13 17 19 23 29 31: Quaternary base Prime factors of the base: 2 Prime factors of one below the base: 3 Prime factors of one above the base: 5 (=11 4) Other prime factors: 13 23 31 101 103 113 131 133: Fraction Prime factors of the denominator: Positional representation Positional ...