**Some Values of The Function**

The first 99 values (sequence A000010 in OEIS) are shown in the table and graph below:

+0 | +1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 | |
---|---|---|---|---|---|---|---|---|---|---|

0+ | 1 | 1 | 2 | 2 | 4 | 2 | 6 | 4 | 6 | |

10+ | 4 | 10 | 4 | 12 | 6 | 8 | 8 | 16 | 6 | 18 |

20+ | 8 | 12 | 10 | 22 | 8 | 20 | 12 | 18 | 12 | 28 |

30+ | 8 | 30 | 16 | 20 | 16 | 24 | 12 | 36 | 18 | 24 |

40+ | 16 | 40 | 12 | 42 | 20 | 24 | 22 | 46 | 16 | 42 |

50+ | 20 | 32 | 24 | 52 | 18 | 40 | 24 | 36 | 28 | 58 |

60+ | 16 | 60 | 30 | 36 | 32 | 48 | 20 | 66 | 32 | 44 |

70+ | 24 | 70 | 24 | 72 | 36 | 40 | 36 | 60 | 24 | 78 |

80+ | 32 | 54 | 40 | 82 | 24 | 64 | 42 | 56 | 40 | 88 |

90+ | 24 | 72 | 44 | 60 | 46 | 72 | 32 | 96 | 42 | 60 |

The top line, *y* = *n* − 1, is a true upper bound. It is attained whenever *n* is prime.

The lower line, *y* ≈ 0.267*n* which connects the points for *n* = 30, 60, and 90 is misleading. If the plot were continued, there would be points below it.

(Examples: for *n* = 210 = 7×30, φ(*n*) ≈ 0.229 *n*; for *n* = 2310 = 11×210 φ(*n*) ≈ 0.208 *n*; and for *n* = 30030 = 13×2310 φ(*n*) ≈ 0.192 *n*.)

In fact, there is no lower bound straight line; no matter how gentle the slope of a line (through the origin) is, there will eventually be points of the plot below the line.

Read more about this topic: Euler's Totient Function

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