FormalConjecturesForMathlib.NumberTheory.PracticalNumbers

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module


public import FormalConjecturesForMathlib.NumberTheory.AdditivelyComplete
public import Mathlib.NumberTheory.Divisors

@[expose] public section

Practical numbers

A positive integer $n$ is called a practical number if every positive integer $m ≤ n$ can be represented as a sum of distinct divisors of $n$.

References:

Wikipedia, Practical number

OEIS A005153

Main definitions

Nat.IsPractical: A natural number $n$ is practical if every positive integer $m ≤ n$ can be expressed as a sum of distinct divisors of $n$.

namespace Nat

A natural number $n$ is practical if every positive integer $m ≤ n$ can be represented as a sum of distinct divisors of $n$.

def IsPractical (n : ℕ) : Prop :=
  ∀ m, m ≤ n → m ∈ subsetSums n.divisors

end Nat