The Main Functions

[x], the largest integer ≤ x

B(n) =

pα n

αp, the sum of the prime factors of n with multiplicity

B1(n) =

pα 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

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