Deficient Number Test

Most numbers are miserly, not generous: their proper divisors add to less than the number itself. Eight is typical - 1 + 2 + 4 = 7 - and nearly three quarters of all integers behave this way. Primes are all deficient (their divisors sum to 1), powers of two nearly perfect (16 misses 16 by one: sigma is 31), and even 945, the famous odd abundant, sits inside a sea of deficients.

Deficient 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 - deficient, perfect or abundant - with factorization, proper-divisor sum and the deficit 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: Deficient, perfect or abundant, decided by the sigma = 2n comparison.
• Deficit shown: Every deficient row carries the amount by which divisors fall short.
• Structure-aware: Prime and power-of-two diagnoses ride along with every verdict.
• 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 deficients, the prime/power-of-two laws, the three-quarter density.
• 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 - deficient, perfect, or abundant.
3. See the sigma arithmetic - sigma, proper-divisor sum and the deficit or excess.
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 rule of the majority. Enter 8 and 10. 8 misses by one: proper divisors 7 against 8. 10 misses by three: divisors 1 + 2 + 5 = 8.

Example 2 - The primes. Enter 41 any prime: the proper sum is always 1, deficit is always p - 1 - the simplest verdict in number theory.

Example 3 - One short of perfect. Enter 2048. sigma(2048) = 4095, proper sum 2047 - short of the number by exactly one, the hallmark of a power of two.

Example 4 - The borderlines. Enter 28 and 12. 28 is perfect; 12 is abundant. Same comparison, three verdicts - the badge shows which side each number falls on.

Example 5 - Batch audit. Paste 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19 together: sixteen consecutive deficients, closing out the opening run of the list.

• Proven verdicts: Every classification rests on exact sigma arithmetic, not lookup tables.
• Deficit in numbers: The shortfall amount makes the deficient condition concrete on every row.
• Structure-aware rows: Prime and nearly perfect power-of-two cases carry their own explaining detail.
• Threshold clarity: Perfect and abundant 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.