Abundancy Ratio
Compute the abundancy index σ(n)/n over all positive divisors, returning a reduced exact fraction and its floating-point value; trial division is limited to n≤10¹².
Description
The abundancy index compares the sum of all positive divisors with the original integer.
The abundancy index compares the sum of all positive divisors with the original integer. A value of two marks a perfect number; values above two indicate abundant numbers.1
The tool returns numerator, denominator, value. The indexing and output conventions below determine how to read these values.
When to use Abundancy Ratio
Use this for the following task: Compute the abundancy index σ(n)/n over all positive divisors, returning a reduced exact fraction and its floating-point value; trial division is limited to n≤10¹². Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Abundancy Ratio and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter n according to the field table, then run the tool.
- Read numerator, denominator, value. Preserve exact decimal strings and the stated indexing or coefficient order.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| n | Positive integer (trial-division resource limit 10^12) | integer; minimum 1; maximum 1000000000000 | Required |
Use factorization to calculate the exact divisor sum, reduce its ratio to n by gcd, and return numerator and denominator strings alongside a floating-point quotient.
abundancy = σ(n)/n; σ(n)=Π[p^a∥n](p^(a+1)−1)/(p−1) The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| numerator | Reduced sum-of-divisors numerator | string (integer/arbitrary) |
| denominator | Reduced integer denominator | string (integer/arbitrary) |
| value | Abundancy index | number |
{
"input": {
"n": 12
},
"output": {
"numerator": "7",
"denominator": "3",
"value": 2.3333333333333335
}
}Limitations and assumptions
The input is at most 10¹². The floating value is approximate; the reduced integer fraction preserves the exact ratio. Proper-divisor sum alone is not the numerator.
Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.
Alternative or Complementary methods
Choose Sum of Squared Divisors when you need its operation: Return σ₂(n), the exact sum of squares of all positive divisors, using prime-power products and trial division limited to n≤10¹². Choose Practical Number Checker for the complementary operation: Check whether every integer from one through n is a sum of distinct divisors of n using the Stewart–Sierpiński prime-factor criterion; trial division is limited to n≤10¹².
Background reference: Measuring Abundance with Abundancy Index.
References
Similar or alternative tools
- Sum of Squared Divisors
Return σ₂(n), the exact sum of squares of all positive divisors, using prime-power products and trial division limited to n≤10¹².
- Practical Number Checker
Check whether every integer from one through n is a sum of distinct divisors of n using the Stewart–Sierpiński prime-factor criterion; trial division is limited to n≤10¹².