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Everything about Practical number

Number whose sums of distinct divisors represent all smaller numbers

In number theory, a practical number or panarithmic number is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Practical numberA 5-minute orientation
  2. Skim a recent paper: A high-performance hybrid random number generator based on a nondegenerate coupled chaos and its practical implementationInfluential work since 2022
  3. Pick up a classic book: Step-By-Step Medical CodingThe most-reprinted book on the subject
  4. Read the landmark paper: Beating the Photon-Number-Splitting Attack in Practical Quantum CryptographyCited 1,376 times
  5. Go sideways: Giuseppe MelfiA closely related topic
  6. Go sideways: List of number theory topicsA closely related topic

📖 Definitions

Noun Any positive integer having the property that all smaller integers can be represented as sums of distinct divisors of it

🗂️ The subject at a glance

  1. Characterization of practical numbers
  2. Properties
  3. Relation to other classes of numbers
  4. Practical numbers and Egyptian fractions
  5. Analogies with prime numbers
  6. The number of prime factors, the number of divisors, and the sum of divisors

🏆 Landmark papers (most cited)

🆕 Influential since 2022

📚 Books worth reading

Ranked by number of editions on Open Library, a rough measure of how lasting a book is.

🔗 Related topics