How to use it
Type a whole number, for example 28. All properties appear at once. Divisors are listed when there are at most 100 of them; their count and sum are always shown. The triangular and Fibonacci rows show the index (T7 means the 7th triangular number, F10 the 10th Fibonacci number).
How it works
The number is factored by trial division. From the factorization n = p₁^e₁ · p₂^e₂ · … come the number of divisors (e₁ + 1)(e₂ + 1)…, the divisor sum σ(n) = ∏ (p^(e+1) − 1)/(p − 1) and Euler’s totient φ(n) = n · ∏ (1 − 1/p). Comparing σ(n) − n with n gives perfect, abundant or deficient. A square or cube is checked with an integer root, a triangular number by testing whether 8n + 1 is a perfect square, and a Fibonacci number by walking the sequence.
Worked example
n = 28 = 2² × 7. It has (2 + 1)(1 + 1) = 6 divisors: 1, 2, 4, 7, 14, 28. Their sum is σ(28) = 7 · 8 = 56, so the proper divisors add up to 56 − 28 = 28: 28 is perfect. It is even, not a square or a cube, triangular (T7 = 7 · 8/2), not a Fibonacci number, its digit sum is 10, φ(28) = 12 and in binary it is 11100.
Tips and common mistakes
- 1 is neither prime nor composite; it has a single divisor.
- Large numbers with many small factors can have thousands of divisors; then only the count and sum are shown.
- All even perfect numbers have the form 2^(p−1)(2^p − 1) with 2^p − 1 prime; no odd perfect number is known.