Rough Number Count Approximation
ExactfiniteBrough(all prime factors≥B) count1≤m≤N, included1by empty factor convention, N≤1e5. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
A B-rough integer has no prime factor below B, with one included by the empty-factor convention.
A B-rough integer has no prime factor below B, with one included by the empty-factor convention.1
This tool counts qualifying integers exactly in a bounded range despite its approximation title.
When to use Rough Number Count Approximation
This tool counts qualifying integers exactly in a bounded range despite its approximation title.
How to use Rough Number Count Approximation and how it works
- Prepare cutoff, prime factor boundary, using the input definitions and matching units below.
- Enter the required fields and run Rough Number Count Approximation.
- Read result.exactObservedCount 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; finite; other constraints below |
| actualPrimeFactorBoundary | Prime-factor threshold B; equality is included on the named smooth or rough branch. | number | Required; finite; other constraints below |
Require cutoff N integer 1–100000 and factor boundary B integer 2–100000. Sieve primes through N and record each integer's largest and smallest prime factors. Include one by empty-factor convention. Count integers 2–N with largest factor<=B for smooth, or smallest factor>=B for rough. Return an exact finite count, not an asymptotic estimate; equality at B is included.
smoothCount=1+count(2<=m<=N:largestPrimeFactor(m)<=B); roughCount=1+count(2<=m<=N:smallestPrimeFactor(m)>=B) 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.exactObservedCount | Exact finite smooth or rough count, including one. | number |
{
"input": {
"actualCutoff": 10,
"actualPrimeFactorBoundary": 3
},
"output": {
"result": {
"exactObservedCount": 5
}
}
}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 Semiprime Count Approximation for a complementary operation: A semiprime is a product of two primes, including a repeated prime factor.
References
-
Rough Number 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.
- Semiprime Count Approximation
A semiprime is a product of two primes, including a repeated prime factor.