Almost Perfect Numbers Generator

Almost perfect numbers are the numbers that miss perfection by exactly one: the sum of all divisors of n equals 2n − 1. Every power of two passes the test — 1, 2, 4, 8, 16, 32, 64, ... — because the divisors of 2^m sum to 2^(m+1) − 1 per the geometric series. Whether any other number passes remains one of the quiet open problems of elementary number theory.

Almost Perfect Numbers Generator walks the powers of two term by term, verifying sigma(n) = 2n − 1 for each exponent, and lets you inspect values with hundreds of digits computed exactly in BigInt.

• Every row self-verifying: The divisor sum sigma(n) is computed and shown beside 2n - 1.
• Exact BigInt values: Powers of two up to 2^2048 (617 digits) rendered without a single error.
• Exponent control: Start from any exponent from 0 and take any window of up to 500 terms.
• Digit counts on rows: The decimal length grows visibly with the exponent.
• Reference tab: The conjecture, the geometric-series proof, and history of the search.
• Free and private: All computation is local; nothing is uploaded, stored or logged.

1. Set the starting exponent — first power of two to list (0 to 2,048).
2. Set the count — consecutive powers to generate (1 to 500).
3. Generate — each row shows n, sigma(n) and the check sigma(n) = 2n - 1.
4. Inspect large rows — 600+ digit values with their digit counts.
5. Copy or download — the run as index = value lines.

Example 1 — The first five. Count 5: 1, 2, 4, 8, 16. For n = 8, sigma(8) = 15 and 2n - 1 = 15: the balance holds exactly.

Example 2 — The geometric proof visible. n = 32: divisors 1+2+4+8+16+32 = 63 = 2 x 32 - 1. Each row displays this tally.

Example 3 — A leap in size. From exponent 40 to 60 the numbers jump from 13 to 19 digits — still exact, still one off from double.

Example 4 — Maximum digits. Exponent 2048 yields a 617-digit value: the largest exactly rendered here, checking sigma = 2n - 1 to the very last digit.

Example 5 — The long view. Start exponent 500, count 12: twelve consecutive powers of two, each with sigma(n) exactly 1 short of double.

• Definition-faithful: Every term comes with its computed sigma(n) check; nothing is assumed.
• True BigInt: Up to 617 digits exact in the browser.
• Flexible windows: Any exponent origin and run length up to 500.
• Clean output: Copy or download plain index = value lines.
• Educational reference: The open problem and its known bounds explained.
• Private by design: No servers, no uploads, no logging.