About

Misidentifying a number's properties leads to downstream errors in cryptography key selection, hash table sizing, and combinatorial proofs. This tool computes over 30 attributes of any integer: prime factorization, the complete divisor set, aliquot sum classification (perfect, abundant, or deficient), digit-level statistics including digital root and digit frequency, membership tests for sequences like Fibonacci and triangular numbers, and base representations in binary, octal, and hexadecimal. It also generates the full Collatz sequence for n and visualizes its trajectory. All computations run client-side with no server dependency.

The tool handles integers up to approximately 9 × 10¹⁵ with exact arithmetic via JavaScript's BigInt. Primality testing uses trial division up to √n for moderate values and falls back to a deterministic Miller-Rabin variant for larger inputs. Note: factorization of semiprimes above 10¹² may take several seconds. Roman numeral output is limited to n ≤ 3,999,999. Pro tip: use the divisor count to verify Euler's totient calculations, or check the digital root as a fast casting-out-nines sanity check on arithmetic results.

Formulas

The primary factorization algorithm decomposes an integer n into its unique prime factors by trial division with a 6k ± 1 wheel optimization, testing candidates up to √n:

n = k∏i=1 p_i^a_i

where each p_i is a distinct prime and a_i ≥ 1 is its multiplicity.

The divisor count τ(n) and divisor sum σ(n) are derived from the factorization:

τ(n) = k∏i=1 (a_i + 1)

σ(n) = k∏i=1 p_i^a_i+1 − 1p_i − 1

Euler's totient function counts integers from 1 to n that are coprime to n:

φ(n) = n k∏i=1 (1 − 1p_i)

The digital root is computed without iteration using modular arithmetic:

dr(n) =

{

0 if n = 09 if n mod 9 = 0n mod 9 otherwise

The Collatz conjecture iterates:

n_i+1 =

{

n_i2 if n_i is even3n_i + 1 if n_i is odd

Where n = input integer, p_i = i-th prime factor, a_i = exponent of p_i, τ = divisor count function, σ = divisor sum function, φ = Euler's totient, dr = digital root.

Reference Data

Property	Definition	Example	Formula / Test
Prime	Divisible only by 1 and itself	29	No divisor in 2..√n
Composite	Has divisors other than 1 and n	28	¬ prime ∧ n > 1
Perfect Number	Equals sum of proper divisors	28 = 1+2+4+7+14	σ(n) − n = n
Abundant	Sum of proper divisors exceeds n	12: 1+2+3+4+6 = 16	σ(n) − n > n
Deficient	Sum of proper divisors less than n	8: 1+2+4 = 7	σ(n) − n < n
Triangular	Sum of first k naturals	21 = 1+2+…+6	8n + 1 is a perfect square
Fibonacci	Member of the Fibonacci sequence	13	5n² ± 4 is a perfect square
Palindrome	Reads the same forwards and backwards	12321	String reversal equality
Armstrong	Sum of digits^k equals n (k = digit count)	153 = 1³+5³+3³	∑ d_i^k = n
Harshad	Divisible by its digit sum	18 ÷ 9 = 2	n mod S(n) = 0
Happy Number	Iterating sum of squared digits reaches 1	19 → 82 → … → 1	Cycle detection (Floyd's)
Perfect Square	√n is an integer	144 = 12²	floor(√n)² = n
Perfect Cube	Cube root of n is an integer	125 = 5³	round(n^1/3)³ = n
Even / Odd	Divisibility by 2	42 (even)	n mod 2
Digital Root	Iterative digit sum until single digit	493 → 16 → 7	1 + (n − 1) mod 9
Digit Sum	Sum of all digits	493 → 16	∑ d_i
Euler's Totient	Count of integers ≤ n coprime to n	φ(12) = 4	n ∏ (1 − 1p)
Collatz Length	Steps to reach 1 via Collatz rule	27 → 111 steps	Iterative: if even n/2, else 3n+1
Roman Numeral	Standard Roman representation	2024 → MMXXIV	Subtractive notation, vinculum for >3999
Binary	Base-2 representation	42 → 101010	n.toString(2)
Octal	Base-8 representation	42 → 52	n.toString(8)
Hexadecimal	Base-16 representation	255 → FF	n.toString(16)

Frequently Asked Questions

Negative integers are accepted. The tool computes the absolute value for factorization, divisor analysis, and base conversions, while preserving the sign for display. Properties like "even/odd" apply to the signed value. Prime testing requires positive integers greater than 1, so negative inputs are classified as "not prime" per standard convention.

The tool handles integers up to approximately 9 × 10¹⁵ (the Number.MAX_SAFE_INTEGER boundary). Prime factorization uses trial division up to √n, so a semiprime near 10¹⁵ requires testing up to ~31.6 million candidates, which may take 2 - 5 seconds. The computation runs in a Web Worker to keep the UI responsive.

The digit sum adds all digits once: for 493, it is 4 + 9 + 3 = 16. The digital root repeats this process until a single digit remains: 16 → 1 + 6 = 7. The shortcut formula is 1 + (n − 1) mod 9, which is equivalent to repeated digit summation and is used as a casting-out-nines check in manual arithmetic.

A positive integer n is a Fibonacci number if and only if at least one of 5n² + 4 or 5n² − 4 is a perfect square. This is the Gessel-Rao identity. It runs in constant time regardless of how large n is, making it far more efficient than iterating through the sequence.

The abundance is σ(n) − 2n. A negative value means deficient (most integers), zero means perfect (only 51 known as of 2024), and positive means abundant. Highly abundant numbers appear frequently in combinatorics and are preferred for hash table sizes because their large divisor count distributes keys more evenly.

The Collatz conjecture (also called the 3n + 1 problem) remains unproven as of 2024. It has been verified computationally for all integers up to approximately 2.95 × 10²⁰. This tool caps the iteration at 10,000 steps. If the sequence has not reached 1 by then, it reports the count as "> 10,000" with a note.