Woodall Number Checker

Check whether a nonnegative decimal integer has the form k·2^k−1 for k≥1, without requiring primality; input is capped at 200 characters.

Description

Woodall numbers form the sequence k·2^k−1 for positive integers k.

Woodall numbers form the sequence k·2^k−1 for positive integers k. Membership concerns this algebraic form; a Woodall number need not be prime.^[1]

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

When to use Woodall Number Checker

Use this for the following task: Check whether a nonnegative decimal integer has the form k·2^k−1 for k≥1, without requiring primality; input is capped at 200 characters. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Woodall 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 isWoodall. 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	Nonnegative decimal integer	string (integer/arbitrary); maximum text length 200; pattern ^(0\|[1-9][0-9]*)$	Required

Read the nonnegative decimal string with BigInt. Generate candidates in increasing k using a shift, stopping at a match or once the candidate exceeds n.

n is Woodall ⇔ n = k×2^k−1 for some k≥1

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

Output field	Meaning	Value type
isWoodall	Whether n is a Woodall number	boolean

json

{
  "input": {
    "n": "63"
  },
  "output": {
    "isWoodall": true
  }
}

Limitations and assumptions

Zero is not a member. Large integers are accepted as strings within the 200-character bound; this is a form check rather than a prime test.

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 Cullen Number Checker when you need its operation: Check whether a nonnegative decimal integer has the form k·2^k+1 for k≥0, including the initial Cullen number one; input is capped at 200 characters. Choose Fermat Number Checker for the complementary operation: Check whether a nonnegative decimal integer has the form 2^(2^k)+1 for k≥0, including composite Fermat numbers; input is capped at 200 characters.

Background reference: OEIS A003261.

References

OEIS A003261 — OEIS Foundation

Similar or alternative tools

Cullen Number Checker
Check whether a nonnegative decimal integer has the form k·2^k+1 for k≥0, including the initial Cullen number one; input is capped at 200 characters.
Fermat Number Checker
Check whether a nonnegative decimal integer has the form 2^(2^k)+1 for k≥0, including composite Fermat numbers; input is capped at 200 characters.