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Friendly Number — Definition, Formula & Examples

A friendly number is a natural number that shares its abundancy index (the ratio of the sum of its divisors to itself) with at least one other natural number. Two numbers that share the same abundancy index form a friendly pair.

A positive integer nn is called friendly if there exists a distinct positive integer mm such that σ(n)n=σ(m)m\frac{\sigma(n)}{n} = \frac{\sigma(m)}{m}, where σ(n)\sigma(n) denotes the sum of all positive divisors of nn. The ratio σ(n)n\frac{\sigma(n)}{n} is called the abundancy index of nn. A number with no such partner is called solitary.

Key Formula

Abundancy index of n=σ(n)n\text{Abundancy index of } n = \frac{\sigma(n)}{n}
Where:
  • nn = A positive integer
  • σ(n)\sigma(n) = The sum of all positive divisors of n

How It Works

To determine whether a number is friendly, compute its abundancy index by finding the sum of all its divisors and dividing by the number itself. Then check whether any other number shares that same ratio. For example, 6 and 28 are both friendly because they share the abundancy index 2 — these are also perfect numbers, since σ(n)=2n\sigma(n) = 2n. Finding friendly pairs can be computationally intensive, and for many small numbers it remains an open question whether they are friendly or solitary.

Worked Example

Problem: Show that 6 and 28 are a friendly pair.
Step 1: Find the divisors of 6 and compute the abundancy index.
σ(6)=1+2+3+6=12,σ(6)6=126=2\sigma(6) = 1 + 2 + 3 + 6 = 12, \quad \frac{\sigma(6)}{6} = \frac{12}{6} = 2
Step 2: Find the divisors of 28 and compute the abundancy index.
σ(28)=1+2+4+7+14+28=56,σ(28)28=5628=2\sigma(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56, \quad \frac{\sigma(28)}{28} = \frac{56}{28} = 2
Step 3: Since both numbers have the same abundancy index of 2, they form a friendly pair.
σ(6)6=σ(28)28=2\frac{\sigma(6)}{6} = \frac{\sigma(28)}{28} = 2
Answer: 6 and 28 are friendly because they share the abundancy index 2. (Both are also perfect numbers.)

Why It Matters

Friendly numbers connect to deeper questions about divisor functions and perfect numbers, topics studied in undergraduate number theory courses. Whether certain small numbers like 10 or 14 are friendly or solitary remains an unsolved problem, illustrating how accessible-sounding questions in number theory can be surprisingly difficult.

Common Mistakes

Mistake: Confusing friendly numbers with amicable numbers.
Correction: Amicable numbers are a pair where each is the sum of the other's proper divisors (e.g., 220 and 284). Friendly numbers share the same abundancy index — a different relationship entirely.