Concepts6

Möbius Function and Inversion

The Möbius function μ(n) is 0 if n has a squared prime factor, otherwise it is (-1)^k where k is the number of distinct prime factors.

#mobius function#mobius inversion#dirichlet convolution+12

∑MathAdvanced

Divisor Function Sums

Summing the divisor function d(i) up to n equals counting lattice points under the hyperbola xy ≤ n, which can be done in O(√n) using floor-division blocks.

#divisor function#euler totient#mobius function+11

∑MathIntermediate

Multiplicative Functions

A multiplicative function is an arithmetic function f with f(mn) = f(m)f(n) whenever gcd(m, n) = 1.

#multiplicative function#dirichlet convolution#mobius function+12

∑MathIntermediate

Euler's Totient Function

Euler's Totient Function φ(n) counts how many integers from 1 to n are coprime with n.

#euler totient#phi function#coprime count+12

∑MathIntermediate

Linear Sieve

The linear sieve builds all primes up to n in O(n) time by ensuring each composite is marked exactly once by its smallest prime factor (SPF).

#linear sieve#smallest prime factor#spf+12

∑MathIntermediate

Sieve of Eratosthenes

The Sieve of Eratosthenes marks multiples of each prime to find all primes up to n in O(n log log n) time.

#sieve of eratosthenes#segmented sieve#linear sieve+11