is the number of integers for which the totient function , also called the multiplicity of (Guy 1994). Erdős (1958) proved that if a multiplicity occurs once, it occurs infinitely often.
The values of for , 2, ... are 2, 3, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, ... (OEIS A014197), and the nonzero values are 2, 3, 4, 4, 5, 2, 6, 6, 4, 5, 2, 10, 2, 2, 7, 8, 9, ... (OEIS A058277), which occur for , 2, 4, 6, 8, 10, 12, 16, 18, 20, ... (OEIS A002202). The table below lists values for .
such that | ||
1 | 2 | 1, 2 |
2 | 3 | 3, 4, 6 |
4 | 4 | 5, 8, 10, 12 |
6 | 4 | 7, 9, 14, 18 |
8 | 5 | 15, 16, 20, 24, 30 |
10 | 2 | 11, 22 |
12 | 6 | 13, 21, 26, 28, 36, 42 |
16 | 6 | 17, 32, 34, 40, 48, 60 |
18 | 4 | 19, 27, 38, 54 |
20 | 5 | 25, 33, 44, 50, 66 |
22 | 2 | 23, 46 |
24 | 10 | 35, 39, 45, 52, 56, 70, 72, 78, 84, 90 |
28 | 2 | 29, 58 |
30 | 2 | 31, 62 |
32 | 7 | 51, 64, 68, 80, 96, 102, 120 |
36 | 8 | 37, 57, 63, 74, 76, 108, 114, 126 |
40 | 9 | 41, 55, 75, 82, 88, 100, 110, 132, 150 |
42 | 4 | 43, 49, 86, 98 |
44 | 3 | 69, 92, 138 |
46 | 2 | 47, 94 |
48 | 11 | 65, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210 |
The smallest such that has exactly 2, 3, 4, ... solutions are given by 1, 2, 4, 8, 12, 32, 36, 40, 24, ... (OEIS A007374). Including Carmichael's conjecture that has no solutions, the smallest such that has exactly 0, 1, 2, 3, 4, ... solutions are given by 3, 0, 1, 2, 4, 8, 12, 32, 36, 40, 24, ... (OEIS A014573). A table listing the first value of with multiplicities up to 100 follows.
0 | 3 | 26 | 2560 | 51 | 4992 | 76 | 21840 |
2 | 1 | 27 | 384 | 52 | 17640 | 77 | 9072 |
3 | 2 | 28 | 288 | 53 | 2016 | 78 | 38640 |
4 | 4 | 29 | 1320 | 54 | 1152 | 79 | 9360 |
5 | 8 | 30 | 3696 | 55 | 6000 | 80 | 81216 |
6 | 12 | 31 | 240 | 56 | 12288 | 81 | 4032 |
7 | 32 | 32 | 768 | 57 | 4752 | 82 | 5280 |
8 | 36 | 33 | 9000 | 58 | 2688 | 83 | 4800 |
9 | 40 | 34 | 432 | 59 | 3024 | 84 | 4608 |
10 | 24 | 35 | 7128 | 60 | 13680 | 85 | 16896 |
11 | 48 | 36 | 4200 | 61 | 9984 | 86 | 3456 |
12 | 160 | 37 | 480 | 62 | 1728 | 87 | 3840 |
13 | 396 | 38 | 576 | 63 | 1920 | 88 | 10800 |
14 | 2268 | 39 | 1296 | 64 | 2400 | 89 | 9504 |
15 | 704 | 40 | 1200 | 65 | 7560 | 90 | 18000 |
16 | 312 | 41 | 15936 | 66 | 2304 | 91 | 23520 |
17 | 72 | 42 | 3312 | 67 | 22848 | 92 | 39936 |
18 | 336 | 43 | 3072 | 68 | 8400 | 93 | 5040 |
19 | 216 | 44 | 3240 | 69 | 29160 | 94 | 26208 |
20 | 936 | 45 | 864 | 70 | 5376 | 95 | 27360 |
21 | 144 | 46 | 3120 | 71 | 3360 | 96 | 6480 |
22 | 624 | 47 | 7344 | 72 | 1440 | 97 | 9216 |
23 | 1056 | 48 | 3888 | 73 | 13248 | 98 | 2880 |
24 | 1760 | 49 | 720 | 74 | 11040 | 99 | 26496 |
25 | 360 | 50 | 1680 | 75 | 27720 | 100 | 34272 |
It is thought that (i.e., the totient valence function never takes on the value 1), but this has not been proven. This assertion is called Carmichael's totient function conjecture and is equivalent to the statement that for all , there exists such that (Ribenboim 1996, pp. 39-40). Any counterexample must have more than digits (Schlafly and Wagon 1994; erroneously given as in Conway and Guy 1996).