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# Prime factors

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In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called prime factorization.

n |

1 |

n |

1 |

## Contents

## Canonical representation

The canonical representation of the unique prime factorization ofn |

ω (n) |

n |

pi |

i |

n |

## Prime factors (with multiplicity)

Factoring a numbern |

pi |

n |

*multiplicity*of

pi |

αi |

pi^{ αi} |

n |

### Number of prime factors (with multiplicity)

The arithmetic functionΩ (n) |

n |

### Sum of prime factors (with multiplicity)

## Distinct prime factors

### Number of distinct prime factors

The arithmetic functionω (n) |

n |

if you forgive the tautological expression.

### Sum of distinct prime factors

## Coprimality

Two positive integers are coprime if and only if they share no common prime factors. The integer1 |

a |

b |

1 |

gcd (a, b) = 1 |

gcd (1, b) = 1 |

b ≥ 1 |

## Sequences

The numberω (n) |

n, n ≥ 1, |

- {0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, ...}

Ω (n) |

n, n ≥ 1, |

- {0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, ...}

n = 1 |

1 |

n, n ≥ 1, |

- {1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, ...}

n = 1 |

1 |

n, n ≥ 1, |

- {1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, ...}

## See also

- Integer factorization
- Divisibility
- Divisibility triangle
- Table of divisors (or table of factors)
- Divisors (or factors)
- Number of divisors (or number of factors)
- Sum of divisors (or sum of factors)

- Prime factorization (or prime factorization (with multiplicity))

- Prime power factorization
- Coprime
- Coprimality
- Coprimality triangle
- Greatest common divisor
- Euclid's algorithm