# prime signature

The prime signature^{} of an integer $n$ is the list of nonzero exponents ${a}_{i}$ from the integer factorization

$$n=\prod _{i=1}^{\mathrm{\infty}}p_{i}{}^{{a}_{i}},$$ |

(with ${p}_{i}$ being the $i$th prime) sorted in ascending order (http://planetmath.org/AscendingOrder) but with duplicates retained. Three examples: the prime signature of 10368 is (4, 7), the prime signature of 10369 is (1), the prime signature of 10370 is (1, 1, 1, 1).

The prime signature of a number is insufficient to uniquely identify it. Numbers like 34992 and 514714375 also have prime signatures of (4, 7). However, prime signatures can identify some kinds of numbers: the primes have signature^{} (1); the squares of primes have signature (2), while other semiprimes have signature (1, 1); sphenic numbers^{} have signature (1, 1, 1); etc. But while other kinds of numbers have different signatures among their members, some generalizations^{} can still be made, such as that highly composite numbers have prime signatures in reverse order of the factorization as usually stated with the primes from 2 up; or that Achilles numbers don’t have any 1s in their prime signature but the greatest common divisor^{} of the numbers in the prime signature is 1.

Title | prime signature |
---|---|

Canonical name | PrimeSignature |

Date of creation | 2013-03-22 18:51:50 |

Last modified on | 2013-03-22 18:51:50 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A41 |