Mobius Function Integer Calculator

Factor n, return zero if any prime exponent exceeds one, otherwise apply minus one to the number of distinct prime factors. The special value mu(1) is one because its factorization has zero distinct primes. Inputs must be whole numbers inside the displayed safe bound. The calculator does not round decimals into the domain or silently substitute a different convention, so verify each entered integer before interpreting the exact result.

The output classifies square-free structure and parity; it is not a primality indicator because many composite values return either sign. A zero result specifically records a squared prime divisor, while either sign records a square-free integer. Use the displayed factors, recurrence step, or modular residue as an independent check. Exact integer output means the implemented arithmetic has no decimal approximation within its bound; it does not prove an unrelated theorem or an unstated combinatorial model.

Trial factorization is capped at one billion and this educational result is not a cryptographic assurance. The implementation deliberately favors transparent factorization over probabilistic methods for very large integers. The bound keeps browser work predictable and every returned integer within JavaScript's exact safe range. Larger inputs need arbitrary-precision software, explicit resource controls, and independent verification rather than treating an overflowed floating-point value as exact.