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Date: 21-10-2019
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Legendre's formula counts the number of positive integers less than or equal to a number which are not divisible by any of the first primes,
(1) |
where is the floor function. Taking , where is the prime counting function, gives
(2) |
Legendre's formula holds since one more than the number of primes in a range equals the number of integers minus the number of composites in the interval.
Legendre's formula satisfies the recurrence relation
(3) |
Let , then
(4) |
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(5) |
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(6) |
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(7) |
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(8) |
where is the totient function, and
(9) |
where . If , then
(10) |
Note that is not practical for computing for large arguments. A more efficient modification is Meissel's formula.
REFERENCES:
Séroul, R. "Legendre's Formula" and "Implementation of Legendre's Formula." §8.7.1 and 8.7.2 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 175-179, 2000.
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