# totient valence function

Given an integer $n$, count how many integers $m$ in the set $\{n+1,n+2,\mathrm{\dots}{n}^{2}\}$ satisfy $\varphi (m)=n$. This is the totient valence of $n$, usually labelled ${N}_{\varphi}(n)$. (The only two special cases are 2 and 6, for which one has to look a little beyound ${n}^{2}$).

Robert Carmichael conjectured that ${N}_{\varphi}(n)=1$ never. Two sequences in Sloane’s OEIS that list numbers with higher totient valences than preceding numbers are A007374 and A097942.

Title | totient valence function^{} |
---|---|

Canonical name | TotientValenceFunction |

Date of creation | 2013-03-22 15:50:57 |

Last modified on | 2013-03-22 15:50:57 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 5 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11A25 |