The Main Functions
[x], the largest integer x
B(n) =
n
αp, the sum of the prime factors of n with multiplicity
B1(n) =
n
pα,
the sum of the largest prime powers dividing n
P (n) = max{p : p|n}, the largest prime factor of the number n 2
p(n) = min{p : p|n}, the smallest prime factor of the number n 2
β(n) =
p|n
p, the sum of the distinct prime factors of n
β∗(n) =
p|n
pP (n)
p = β(n) P (n), the sum of the prime factors of n except for the
largest
π(x) =
p≤x
1, the number of prime numbers x
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
γ(n) =
p|n
p, the product of the prime numbers which divide n
δ(n) =
p n
p, the product of the prime numbers which divide n exactly
φ(n) =
m≤n
(m,n)=1
1, called the Euler φ function, which counts the number of numbers
m n which are co-prime with n
xv
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