Example Data
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| n | Prime factorization | τ(n) | σ(n) | φ(n) | μ(n) | rad(n) |
|---|---|---|---|---|---|---|
| 12 | 22 × 3 | 6 | 28 | 4 | 0 | 6 |
| 28 | 22 × 7 | 6 | 56 | 12 | 0 | 14 |
| 45 | 32 × 5 | 6 | 78 | 24 | 0 | 15 |
| 97 | 97 | 2 | 98 | 96 | -1 | 97 |
| 360 | 23 × 32 × 5 | 24 | 1170 | 96 | 0 | 30 |
Results
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Formulas Used
Let n = ∏ pᵢeᵢ be the prime factorization (pᵢ distinct). Number of divisors: τ(n) = ∏ (eᵢ + 1) Sum of divisors: σ(n) = ∏ (pᵢ^(eᵢ+1) − 1) / (pᵢ − 1) Euler's totient: φ(n) = n ∏ (1 − 1/pᵢ) Möbius function: μ(n) = 0 if any eᵢ ≥ 2; else (−1)^k where k = number of primes Radical: rad(n) = ∏ pᵢ Proper divisor sum: s(n) = σ(n) − n
How to Use This Calculator
- Choose Single integer or Batch range.
- Enter an integer n, or a range like 1 to 100.
- Press Calculate to compute factorization and factor functions.
- Use Download CSV or Download PDF to export results.
- Click Copy shareable link to snapshot inputs in the URL.
Note: Very large n may take longer with trial division.
What are Factor Functions?
Factor functions map an integer to arithmetic properties derived from its prime structure. Examples include τ(n) for the number of divisors, σ(n) for their sum, φ(n) for totatives, μ(n) from multiplicative parity, and rad(n) as product of distinct primes.
These functions are multiplicative for coprime arguments and are central in analytic number theory, divisor-sum identities, and classification of numbers as perfect, abundant, or deficient.
Perfect, Abundant, and Deficient Numbers
Compare s(n) = σ(n) − n to n: if equal, n is perfect (e.g., 6, 28); if larger, abundant (e.g., 12); else deficient (e.g., primes). This tool reports the classification for each single input.