Euler's Totient Function - Some Values of The Function

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 ≈ 0.267n 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

Famous quotes containing the words function and/or values:

    The function of the actor is to make the audience imagine for the moment that real things are happening to real people.
    George Bernard Shaw (1856–1950)

    Normality highly values its normal man. It educates children to lose themselves and to become absurd, and thus to be normal. Normal men have killed perhaps 100,000,000 of their fellow normal men in the last fifty years.
    —R.D. (Ronald David)