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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.
The discovery year and discoverer are of the Mersenne prime, since the perfect number immediately follows by the Euclid–Euler theorem. Discoverers denoted as "GIMPS / name" refer to GIMPS discoveries with hardware used by that person. Later entries are extremely long, so only the first and last six digits of each number are shown.
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An odd prime p has irregular index n if and only if there are n values of k for which p divides B 2k and these ks are less than (p − 1)/2. The first irregular prime with irregular index greater than 1 is 157, which divides B 62 and B 110, so it has an irregular index 2. Clearly, the irregular index of a regular prime is 0.
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 ).
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The multiples of a given prime are generated as a sequence of numbers starting from that prime, with constant difference between them that is equal to that prime. [1] This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime. [ 2 ]