ddidderr
09967ad6d7
Add a concise README that documents how the program finds highly composite numbers. The explanation covers the divisor-count formula, why exponent order lets the search skip arbitrary numbers, and how the implementation generates and filters record candidates. Test Plan: - Not run; documentation-only change. Refs: N/A
60 lines
1.4 KiB
Markdown
60 lines
1.4 KiB
Markdown
# hcn
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Find highly composite numbers up to a search limit.
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```bash
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cargo run --release -- 1000000000
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```
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If no limit is given, the default is `1_000_000_000`.
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## Algorithm
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A number with prime factorization
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```text
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n = 2^a * 3^b * 5^c * ...
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```
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has
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```text
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(a + 1) * (b + 1) * (c + 1) * ...
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```
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divisors.
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The exact primes do not affect the divisor count; only the exponents do.
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For a fixed exponent list, the smallest possible number is made by putting the
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largest exponent on the smallest prime:
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```text
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a >= b >= c >= ...
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```
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If a larger exponent appears on a larger prime, swapping those two exponents
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keeps the divisor count unchanged but makes the number smaller.
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So a highly composite number cannot have that shape. If it did, the smaller
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swapped number would already have the same number of divisors, so the original
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number would not be the first record.
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That means record candidates are only numbers like:
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```text
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2^a * 3^b * 5^c * ... where a >= b >= c >= ...
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```
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All other numbers are redundant for finding new records.
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The program:
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1. Recursively generates numbers from the first primes.
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2. Only tries exponent sequences where each exponent is no larger than the
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previous one.
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3. Stops a branch as soon as the number exceeds the search limit.
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4. Computes the divisor count directly from the exponents.
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5. Sorts candidates by number and prints each new divisor-count record.
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This avoids factoring every number in the range.
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