Composite Density Approximation
Complement of PNT prime density heuristic1−1/lnx, declaredx≥3, not guaranteed count or probability.
Description
A complement of the prime-density heuristic is an asymptotic illustration rather than a measured composite fraction.
A complement of the prime-density heuristic is an asymptotic illustration rather than a measured composite fraction.1
This tool evaluates one minus the reciprocal natural logarithm on a bounded integer domain.
When to use Composite Density Approximation
Use this tool to evaluate one minus the reciprocal natural logarithm on a bounded integer domain.
How to use Composite Density Approximation and how it works
- Prepare actualX, with the units, conventions and assumptions below.
- Enter the required fields and run Composite Density Approximation.
- Read result.value 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 |
|---|---|---|---|
| actualX | Actual X. Use the units and matching context stated in the calculation below. | number | Required; integer 1–1000000; integer 3–1000000; finite; other constraints below |
Require an integer x from 3 to 1,000,000. Return 1-1/log(x), where log is the natural logarithm. This is a prime-density-based composite-density approximation, not an exact count of composite integers up to x.
value=1-1/ln(x) 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.value | Dimensionless composite-density approximation under the stated natural-log formula. | number |
{
"input": {
"actualX": 10
},
"output": {
"result": {
"value": 0.5657055180967483
}
}
}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 Product Approximation for a complementary operation: The zeta Euler product combines factors indexed by primes on a convergent real-exponent domain.
Use Artin Constant Approximation for a complementary operation: Artin's constant has an Euler product over primes related to a primitive-root density convention.
References
-
Composite Density Approximation — tool.io
Similar or alternative tools
- Euler Product Approximation
The zeta Euler product combines factors indexed by primes on a convergent real-exponent domain.
- Artin Constant Approximation
Artin's constant has an Euler product over primes related to a primitive-root density convention.