Abundant Number Test

Twelve is the first number that cannot contain itself: its proper divisors 1, 2, 3, 4 and 6 sum to 16, six more than twelve. Numbers with this overflow are abundant - profligate integers that owe more to their parts than they hold. The list grows fast: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60... and it hides a famous surprise, the odd abundant 945, unnoticed by the ancients for the simple reason that they never looked past seventy.

Abundant Number Test factorizes any value up to a trillion by trial division, computes sigma(n) as the exact product (p^(e+1) - 1)/(p - 1), and returns the verdict from the sigma = 2n comparison - abundant, perfect or deficient - with factorization, proper-divisor sum and the excess amount on every row. Batches of 10,000 run in one pass, all inside the browser.

• Exact sigma: Divisor sums from prime factorizations, never estimated, BigInt throughout.
• Three truthful verdicts: Abundant, perfect or deficient, decided by the sigma = 2n comparison.
• Excess shown: Every abundant row carries the amount by which divisors beat the number.
• Odd cases covered: 945 and all odd abundants classify under the same exact rule.
• Batch testing: Up to 10,000 values per run, each with independent verdict and detail.
• Trillion ceiling: Inputs of up to twelve digits are fully covered by the prime table.
• Copy or download: Verdict list and numbers list, both one-click exports.
• Reference tab: The opening abundances, the 945 story, the abundancy index.
• Free and private: Everything runs locally; nothing is uploaded or stored.

1. Enter numbers - one per line, or a batch separated by commas or spaces.
2. Read the verdict badge - abundant, perfect, or deficient.
3. See the sigma arithmetic - sigma, proper-divisor sum and the excess or deficit.
4. Follow the factorization - the prime split that produces the sum.
5. Copy or download the results for research, teaching or further analysis.

Example 1 - The opener. Enter 12. Divisors sum to 28, proper divisors to 16 - abundant by exactly 4, the first of the list.

Example 2 - The odd pioneer. Enter 945 = 3^3 x 5 x 7. Proper divisors sum to 975, beating 945 by 30 - the smallest odd abundant number, found as exactly as any even one.

Example 3 - The threshold. Enter 20 and 24. 20 is abundant by 2 (divisors 1 + 2 + 4 + 5 + 10 = 22); 24 is abundant by 12 - both rows report their excess.

Example 4 - The borderlines. Enter 28 and 16. 28 is perfect - sigma equals 2n; 16 is deficient, sigma 31 shortfalls to 15. The same comparison yields all three verdicts.

Example 5 - Batch audit. Paste 12, 18, 20, 24, 28, 30, 36, 40, 42, 945 together: nine abundant badges and the perfect 28 in one pass, each with its excess.

• Proven verdicts: Every classification rests on exact sigma arithmetic, not lookup tables.
• Excess in numbers: The overflow amount makes the abundant condition concrete on every row.
• No parity bias: Odd abundants get the same honest treatment as even ones.
• Threshold clarity: Perfect and deficient neighbors appear in the same comparison language.
• Batch speed: Ten thousand candidates classified per run with per-line detail.
• Private and free: Everything runs in the browser; nothing is uploaded, stored or logged.