Deficient Numbers Generator

Sum a number's divisors and most integers come up short: that is deficiency, sigma(n) < 2n. The class opens 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, ... — every prime, every prime power, and roughly three of every four integers in between.

Deficient Numbers Generator builds a divisor-sieve of sigma up to any needed bound, then hands you any window of deficient numbers: start at the 73,000th deficient number and read sigma and deficiency exactly as computed.

• Complete divisor sums: Sigma computed for every integer by a classic near-linear divisor sieve.
• Deficiency on every row: The shortfall 2n - sigma(n) is displayed for each term.
• Start anywhere: Any position from the 1st to the 100,000th deficient number, 1 to 500 consecutive terms.
• Fast at any depth: The sieve makes deep windows as cheap as shallow ones.
• Reference tab: The density theorem, prime-power proofs, and the perfect boundary.
• Free and private: All computation is local; nothing is uploaded, stored or logged.

1. Set the start index — which deficient number to begin with (1 to 100,000).
2. Set the count — consecutive deficient numbers to list (1 to 500).
3. Generate — the sieve fills sigma up to the needed bound instantly.
4. Check sigma and deficiency — the two numbers that certify each entry.
5. Copy or download — the run as index = value lines.

Example 1 — The opening list. Count 14: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16 — most of the smallish integers are deficient.

Example 2 — Unit deficiency. n = 8: sigma = 15 = 2n - 1, deficiency exactly 1; the only integers that come closer to perfection without reaching it are the powers of two.

Example 3 — Prime rows. Count 5 from index 4: 5, 7, 8, 9, 10 — three prime powers (5, 7, 9) plus 8 and 10, each with a visible shortfall.

Example 4 — Deep start. Start index 50,000, count 5: the sieve spans beyond 67,000 to collect the window in milliseconds.

Example 5 — The boundary. Scan the first 20 rows: 12 never appears — it is abundant — and neither does 6, which is perfect. The trichotomy is complete.

• Definition-faithful: Every entry certified by its own sigma count.
• Index-proof speed: Near-linear sieve keeps any window immediate.
• Flexible ranges: Up to 100,000 positions deep, 500 terms wide.
• Clean output: Copy or download index = value lines.
• Educational reference: Density, prime powers, and the perfect boundary.
• Private by design: No servers, no uploads, no logging.