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The multiplicity of a prime which does not divide n may be called 0 or may be considered undefined. 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.
105 is a triangular number, a dodecagonal number, [1] and the first Zeisel number. [2] It is the first odd sphenic number and is the product of three consecutive prime numbers. 105 is the double factorial of 7. [3] It is also the sum of the first five square pyramidal numbers. 105 comes in the middle of the prime quadruplet (101, 103, 107, 109 ...
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a ...
For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(10 1000) ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(10 2000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N). [3]
All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld.
Formula for primes. In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. Formulas for calculating primes do exist; however, they are computationally very slow. A number of constraints are known, showing what such a "formula" can and cannot be.
Prime power. In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: 7 = 71, 9 = 32 and 64 = 26 are prime powers, while 6 = 2 × 3, 12 = 22 × 3 and 36 = 62 = 22 × 32 are not. The sequence of prime powers begins:
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 22 − 1. [1][2] The numbers p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 211 − 1 = 2047 = 23 × 89. [3] Meanwhile, perfect numbers are natural numbers that equal ...