Divisor Sum Approximation

Exact finite sumσ1(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

A sum of divisor sums weights each divisor by its value across the finite prefix.

A sum of divisor sums weights each divisor by its value across the finite prefix.^[1]

This tool computes that weighted total exactly despite the approximation wording in its title.

When to use Divisor Sum Approximation

Use this tool to compute that weighted total exactly despite the approximation wording in its title.

How to use Divisor Sum Approximation and how it works

Prepare actualCutoff, with the units, conventions and assumptions below.
Enter the required fields and run Divisor Sum Approximation.
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. Accumulate d*floor(N/d) with BigInt for d=1 through N and return an exact decimal string, including "0" at N=0. This sums the divisor-sum function over the finite range rather than approximating an infinite series.

value=sum_{d=1..N} d*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

json

{
  "input": {
    "actualCutoff": 5
  },
  "output": {
    "result": {
      "exactDivisorSummatory": "21"
    }
  }
}

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 Summatory Calculator for a complementary operation: The divisor-count summatory function totals the number of positive divisors across integers through a cutoff.

References

Divisor Sum Approximation — 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 Summatory Calculator
The divisor-count summatory function totals the number of positive divisors across integers through a cutoff.