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
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