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 xv

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