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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.
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 ).
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.
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 ]
For example, the odd integer 47 can be expressed as the sum of a prime and a semiprime in four different ways: 47 = 13 + 2×17 = 37 + 2×5 = 41 + 2×3 = 43 + 2×2. The number of ways this can be done is given by OEIS sequence A046927 (Number of ways to express 2n+1 as p+2q where p and q are primes). Lemoine's conjecture is that this sequence ...
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Times style is to always capitalize the first letter of a clue, regardless of whether the clue is a complete sentence or whether the first word is a proper noun. On occasion, this is used to deliberately create difficulties for the solver; e.g., in the clue [John, for one], it is ambiguous whether the clue is referring to the proper name John ...