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.

Description

Cullen numbers have the form k·2^k+1 for nonnegative k.

Cullen numbers have the form k·2^k+1 for nonnegative k. The initial value one is included, and membership does not require a prime value.^[1]

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

When to use Cullen Number Checker

Use this for the following task: 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. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Cullen 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 isCullen. 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

Parse the nonnegative decimal input with BigInt and generate increasing candidates from k=0 until a match or overshoot.

Cullen ⇔ n=k×2^k+1 for some k≥0

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

Output field	Meaning	Value type
isCullen	Whether n is a Cullen number	boolean

json

{
  "input": {
    "n": "65"
  },
  "output": {
    "isCullen": true
  }
}

Limitations and assumptions

Input text is at most 200 characters. Zero does not qualify, and no primality test or sequence list is returned.

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 Woodall Number Checker when you need its operation: 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. Choose Proth Number Checker for the complementary operation: Check whether a decimal integer is k·2^m+1 with m≥1 and positive odd k<2^m, without requiring primality; input is capped at 200 characters.

Background reference: OEIS A002064.

References

OEIS A002064 — OEIS Foundation

Similar or alternative tools

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.
Proth Number Checker
Check whether a decimal integer is k·2^m+1 with m≥1 and positive odd k<2^m, without requiring primality; input is capped at 200 characters.