Staple. All other prior entries in this table were also verified as part of that work. The value for 1027 was announced in 2015 by David Baugh and Kim Walisch. The value for 1028 was announced in 2020 by David Baugh and Kim Walisch. The value for 1029 was announced in 2022 by David Baugh and Kim Walisch. Algorithms for evaluating (x) A simple way to find , if is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to and then to count them. A more elaborate way of finding is due to Legendre (using the inclusion–exclusion principle): given , if are distinct prime numbers, then the number of integers less than or equal to which are divisible by no is (where denotes the floor function). This number is therefore equal to when the numbers are the prime numbers less than or equal to the square root of . The Meissel–Lehmer algorithm In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating Let be the first primes and denote by the number of natural numbers not greater than which are divisible by none of the for any Then Given a natural number if and if then Using this approach, Meissel computed for equal to 5×, , , and . In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real and for natural numbers and as the number of numbers not greater than with exactly prime factors, all greater than Furthermore, set Then where the sum actually has only finitely many nonzero terms. Let denote an integer such that and set Then and when Therefore, The computation of can be obtained this way: where the sum is over prime numbers. On the other hand, the computation of can be done using the following rules: Using his method and an IBM 701, Lehmer was able to compute the correct value of and missed the correct value of by 1. Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat. Other prime-counting functions Other prime-counting functions are also used because they are more convenient to work with. Riemann's prime-power counting function Riemann's prime-power counting function is usually denoted as or It has jumps of at prime powers and it takes a value halfway between the