The Joy of Harshad Numbers - Numberphile

Key Concepts:

• Harshad Number: A number divisible by the sum of its digits.
• Joygiver: The meaning of "Harshad" in Sanskrit.
• Factorial: The product of all positive integers less than or equal to a given number (e.g., 5! = 1 * 2 * 3 * 4 * 5 = 120).
• Harshad Pair/Triple/Quadruple: Consecutive numbers that are all Harshad numbers.
• Density of Harshad Numbers: The frequency of Harshad numbers within a given range.
• Modulo (mod): The remainder after division.
• Harshad Chain: A sequence of numbers where each number, when divided by the sum of its digits, results in another Harshad number.
• Trans-Harshad Number: A number that is Harshad in every base.
• Super Harshad Number (Moran Number): A Harshad number where the result of dividing the number by the sum of its digits is a prime number.
• Harshadmorphic Number: A Harshad number that ends in a specific number, and the sum of its digits is equal to that specific number.

1. Definition and Examples of Harshad Numbers

• A Harshad number is defined as any number that is divisible by the sum of its digits.
• The term "Harshad" comes from Sanskrit, meaning "joygiver."
• Examples provided include:
  • 2025: 2 + 0 + 2 + 5 = 9, and 2025 / 9 = 225 (a whole number).
  • 666: 6 + 6 + 6 = 18, and 666 / 18 = 37 (a whole number).
  • 1729 (the "taxi cab number"): 1 + 7 + 2 + 9 = 19, and 1729 / 19 = 91 (a whole number).

2. Historical Context and Notable Mathematicians

• Harshad numbers were originally introduced by D.R. Kaprekar.
• They were later brought to prominence by mathematician Ivan Niven in the 1970s.

3. Properties and Patterns of Harshad Numbers

• Factorials: Factorials are often Harshad numbers due to having many factors. However, not all factorials are Harshad (e.g., 432! is the first factorial that is not Harshad).
• Single-Digit Numbers: Every single-digit number is a Harshad number.
• Infinite Number: There are an infinite number of Harshad numbers, but their density decreases as numbers get larger.
• Consecutive Harshad Numbers:
  • Harshad pairs exist (e.g., 20 and 21).
  • Harshad triples exist (e.g., 110, 111, and 112).
  • Blocks of up to 20 consecutive Harshad numbers exist.
  • It is proven that there are no blocks of 21 consecutive Harshad numbers.
• Clustering Around Multiples of Nine: Harshad numbers tend to cluster around multiples of nine.
  • This is because any number is equal to the sum of its digits modulo 9.
  • Example: 1729 = (192 * 9) + 1, so 1729 mod 9 = 1. The sum of its digits (19) = (2 * 9) + 1, so 19 mod 9 = 1.

4. Harshad Chains

• A Harshad chain is a sequence where a Harshad number, when divided by the sum of its digits, results in another Harshad number, and so on.
• Example:
  • 6804 / (6 + 8 + 0 + 4) = 6804 / 18 = 378
  • 378 / (3 + 7 + 8) = 378 / 18 = 21
  • 21 / (2 + 1) = 21 / 3 = 7
  • 7 / 7 = 1
• A large number (20165028585798884466176) is mentioned as an example that yields a chain of 12 Harshad numbers.
• It is possible to create infinitely long chains.

5. Harshad Numbers in Different Bases

• The concept of Harshad numbers can be extended to different number bases.
• The presenter uses Python code to check if Brady's birthday is a Harshad number in different bases.
• Trans-Harshad Numbers: Numbers that are Harshad in every base. The only trans-Harshad numbers are 1, 2, 4, and 6.
• The number 12 is Harshad in every base except base 8.

6. Universal Properties and Theorems

• Any number under a billion is either a Harshad number or the sum of two Harshad numbers.
• Every positive integer can be expressed as the sum of at most k Harshad numbers, where k is a finite number. The exact value of k is currently unknown.
  • The proof of this relies on concepts related to Dedekind functions.

7. Super Harshad Numbers (Moran Numbers)

• A Harshad number where the result of dividing the number by the sum of its digits is a prime number.
• Example: 18 / (1 + 8) = 18 / 9 = 2 (2 is a prime number).

8. Harshadmorphic Numbers

• A Harshad number that ends in a specific number, and the sum of its digits is equal to that specific number.
• Examples:
  • 16218: Ends in 18, and 1 + 6 + 2 + 1 + 8 = 18.
  • 2715: Ends in 15, and 2 + 7 + 1 + 5 = 15.
• The number 11 is the only number that is not Harshadmorphic in base 10.

9. Conclusion

The video explores the properties and patterns of Harshad numbers, demonstrating their recreational and mathematical significance. It covers various aspects, including their definition, historical context, distribution, behavior in different bases, and related concepts like Harshad chains, super Harshad numbers, and Harshadmorphic numbers. The video highlights both known facts and unsolved problems related to these "joygiving" numbers.