Concepts6

Harmonic Lemma

The Harmonic Lemma says that the values of \lfloor n/i \rfloor only change about 2\sqrt{n} times, so you can iterate those value blocks in O(\sqrt{n}) instead of O(n).

#harmonic lemma#integer division trick#block decomposition+12

∑MathIntermediate

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle (IEP) corrects overcounting by alternately adding and subtracting sizes of intersections of sets.

#inclusion-exclusion#derangements#surjections+12

∑MathAdvanced

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