Divisor Summatory Calculator
Exact finite sumτ(n) divisor-summatory rearrangement, not single divisor-sum or count. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
The divisor-count summatory function totals the number of positive divisors across integers through a cutoff.
The divisor-count summatory function totals the number of positive divisors across integers through a cutoff.1
This tool computes the finite total exactly by summing floor division counts.
When to use Divisor Summatory Calculator
Use this tool to compute the finite total exactly by summing floor division counts.
How to use Divisor Summatory Calculator and how it works
- Prepare actualCutoff, with the units, conventions and assumptions below.
- Enter the required fields and run Divisor Summatory Calculator.
- Read result.exactDivisorSummatory using the output definitions below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualCutoff | Inclusive integer cutoff N for the finite operation; use the stated tool-specific range. | number | Required; integer 0–1000000; finite; other constraints below |
Require N as an integer from 0 to 1,000,000. Sum floor(N/d) for d=1 through N using BigInt and return a decimal string. N=0 yields "0". This counts the total divisors of the integers through N exactly and does not use an asymptotic approximation.
value=sum_{d=1..N} floor(N/d), encoded as decimal string Apply the input ordering, units and branch conditions stated above.
| Output field | Meaning | Value type |
|---|---|---|
| result | Object containing the calculated or selected fields below. | object |
| result.exactDivisorSummatory | Exact finite divisor total under this tool's weighting, as a decimal integer string. | string |
{
"input": {
"actualCutoff": 5
},
"output": {
"result": {
"exactDivisorSummatory": "10"
}
}
}Limitations and assumptions
The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.
All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.
Alternative or Complementary methods
Use Euler Totient Sum Approximation for a complementary operation: The totient summatory function adds the counts of positive integers coprime to each index through a finite cutoff.
Use Divisor Sum Approximation for a complementary operation: A sum of divisor sums weights each divisor by its value across the finite prefix.
References
-
Divisor Summatory Calculator — tool.io
Similar or alternative tools
- Euler Totient Sum Approximation
The totient summatory function adds the counts of positive integers coprime to each index through a finite cutoff.
- Divisor Sum Approximation
A sum of divisor sums weights each divisor by its value across the finite prefix.