Practical Number Checker

Check whether every integer from one through n is a sum of distinct divisors of n using the Stewart–Sierpiński prime-factor criterion; trial division is limited to n≤10¹².

Description

A practical number can express every integer from one through n as a sum of distinct positive divisors of n.

A practical number can express every integer from one through n as a sum of distinct positive divisors of n. A prime-factor criterion avoids enumerating all those subset sums.^[1]

The tool returns isPractical. The indexing and output conventions below determine how to read these values.

When to use Practical Number Checker

Use this for the following task: Check whether every integer from one through n is a sum of distinct divisors of n using the Stewart–Sierpiński prime-factor criterion; trial division is limited to n≤10¹². Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Practical Number Checker and understand its results

Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
Enter n according to the field table, then run the tool.
Read isPractical. A true flag means the candidate meets the stated property; false means it does not.
If the flag is false, review the candidate and definition. A well-formed nonmember is different from a rejected input.

Input field	Meaning	Value type and bounds	Requirement
n	Positive integer (trial-division resource limit 10^12)	integer; minimum 1; maximum 1000000000000	Required

Factor n in increasing prime order. Start the divisor-sum accumulator at one, reject a prime above accumulator+1, and update the accumulator with its geometric divisor sum.

for each ascending prime p: p≤1+σ(product of preceding prime powers)

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
isPractical	Whether n is practical	boolean

json

{
  "input": {
    "n": 30
  },
  "output": {
    "isPractical": true
  }
}

Limitations and assumptions

The input is at most 10¹². One passes, and the tool returns no particular divisor-sum representations.

Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.

Alternative or Complementary methods

Choose Abundancy Ratio when you need its operation: Compute the abundancy index σ(n)/n over all positive divisors, returning a reduced exact fraction and its floating-point value; trial division is limited to n≤10¹². Choose Distinct Prime Factor Count for the complementary operation: Return ω(n), the number of distinct prime factors, with ω(1)=0 and trial division limited to n≤10¹².

Background reference: OEIS A005153.

References

OEIS A005153 — OEIS Foundation

Similar or alternative tools

Abundancy Ratio
Compute the abundancy index σ(n)/n over all positive divisors, returning a reduced exact fraction and its floating-point value; trial division is limited to n≤10¹².
Distinct Prime Factor Count
Return ω(n), the number of distinct prime factors, with ω(1)=0 and trial division limited to n≤10¹².