Frequently Used
Functions
P (n) = max{p : p|n}, the largest prime factor of n ≥ 2
p(n) = min{p : p|n}, the smallest prime factor of n ≥ 2
Ψ(x, y) = #{n ≤ x : P (n) ≤ y}
Φ(x, y) = #{n ≤ x : p(n) y}
β(n) =
p|n
p, the sum of the prime factors of n
π(x) =
p≤x
1, the number of prime numbers ≤ x
θ(x) =
p≤x
log p
ψ(x) =
pα≤x
α≥1
log p
π(x; k, ) =
p≤x
p≡ (mod k)
1, the number of prime numbers p ≤ x, p ≡ (mod k)
li(x) =
x
0
dt
log t
, the logarithmic integral
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