Semiprime Count Approximation
Exactfinitecountsemiprimespq≤Nincludingp=q, not invalid asymptotic at smallN. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
A semiprime is a product of two primes, including a repeated prime factor.
A semiprime is a product of two primes, including a repeated prime factor.1
This tool counts semiprimes exactly through a supplied finite cutoff rather than using an asymptotic count.
When to use Semiprime Count Approximation
This tool counts semiprimes exactly through a supplied finite cutoff rather than using an asymptotic count.
How to use Semiprime Count Approximation and how it works
- Prepare cutoff, using the input definitions and matching units below.
- Enter the required fields and run Semiprime Count Approximation.
- Read result.exactObservedSemiprimeCount using the units, signs and ordering explained 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 upper integer cutoff for the finite counting or discrepancy operation. | number | Required; integer 1–100000; finite; other constraints below |
Require cutoff integer 1–100000. Sieve primes through cutoff and count products p_i*p_j<=cutoff with j>=i. Including equal indices counts prime squares, and the ordering avoids duplicate products. Return the exact finite semiprime count including multiplicity-two factorizations, not a prime-square-free count.
count=sum(1 where i<=j and prime_i*prime_j<=N) 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.exactObservedSemiprimeCount | Exact finite semiprime count, including prime squares. | number |
{
"input": {
"actualCutoff": 10
},
"output": {
"result": {
"exactObservedSemiprimeCount": 4
}
}
}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 Smooth Number Count Approximation for a complementary operation: A B-smooth integer has no prime factor larger than B, with one included by the empty-factor convention.
Use Rough Number Count Approximation for a complementary operation: A B-rough integer has no prime factor below B, with one included by the empty-factor convention.
References
-
Semiprime Count Approximation — tool.io
Similar or alternative tools
- Smooth Number Count Approximation
A B-smooth integer has no prime factor larger than B, with one included by the empty-factor convention.
- Rough Number Count Approximation
A B-rough integer has no prime factor below B, with one included by the empty-factor convention.