Abundant Numbers Generator

Count everything that divides a number, add it all up, and a minority of integers come out ahead: this is abundance, sigma(n) > 2n. The list opens 12, 18, 20, 24, 30, 36, 40, 42, 48, ... — numbers whose divisors weigh more than the number itself.

Abundant Numbers Generator builds the full divisor-sum table with a sieve, then hands you any window of abundant numbers you ask for, each row carrying sigma(n) and the abundance excess. Start at the 57,000th abundant number if you like — the sieve makes every position equally fast.

• Complete divisor sums: Sigma computed for every integer under the hood by a classic divisor sieve.
• Abundance on every row: The excess sigma(n) - 2n is computed and displayed.
• Start anywhere: Any position from the 1st to the 100,000th abundant number, 1 to 500 consecutive terms.
• Fast by design: The sieve is near-linear, so deep windows cost no extra time.
• Reference tab: The density theorem, 945 and the odd story, primitive abundant numbers.
• Free and private: All computation is local; nothing is uploaded, stored or logged.

1. Set the start index — which abundant number to begin with (1 to 100,000).
2. Set the count — consecutive abundant numbers to list (1 to 500).
3. Generate — a divisor sieve fills sigma up to the needed bound and the window is extracted.
4. Read sigma and abundance — the two numbers that certify each entry.
5. Copy or download — the run as index = value lines.

Example 1 — The opening list. Count 12: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60 — 12 leads with abundance 4.

Example 2 — The first odd. Jump to the 39th abundant number: it is 945 = 3^3 x 5 x 7, the smallest odd one, with abundance 30.

Example 3 — Small abundance. Count 3 from the start: 20 carries abundance 2, the smallest surplus possible (four divisors out of five are proper).

Example 4 — Deep start. Start at index 1,000, count 5: the sieve fills up to around 4,000 candidates in milliseconds and serves the window exactly.

Example 5 — Abundance growth. Start 10,000, count 10: abundance now runs in the hundreds, roughly proportional to the numbers themselves.

• Definition-faithful: Every term certified by its own sigma count.
• Index-proof speed: Near-linear sieve makes any window equal-cost.
• Flexible ranges: Up to 100,000 positions deep, 500 terms wide.
• Clean output: Copy or download index = value lines for analysis.
• Educational reference: Density, primitive abundants, and the odd-number frontier explained.
• Private by design: No servers, no uploads, no logging.