Smooth Number Count Approximation

ExactfiniteBsmooth(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-smooth integer has no prime factor larger than B, with one included by the empty-factor convention.

A B-smooth integer has no prime factor larger than B, with one included by the empty-factor convention.^[1]

This tool counts such integers exactly within a bounded finite range despite the approximation wording in its title.

When to use Smooth Number Count Approximation

This tool counts such integers exactly within a bounded finite range despite the approximation wording in its title.

How to use Smooth 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 Smooth 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

json

{
  "input": {
    "actualCutoff": 10,
    "actualPrimeFactorBoundary": 3
  },
  "output": {
    "result": {
      "exactObservedCount": 7
    }
  }
}

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 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.

Use Semiprime Count Approximation for a complementary operation: A semiprime is a product of two primes, including a repeated prime factor.

References

Smooth Number Count Approximation — tool.io

Similar or alternative tools

Rough Number Count Approximation
A B-rough integer has no prime factor below 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.