is the number of Integers for which , also called the Multiplicity of (Guy 1994). The table below lists values for .
multiplicity | ||
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 |
A table listing the first value of with multiplicities up to 100 follows (Sloane's A014573).
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 10,000,000 Digits (Schlafly and Wagon 1994, Conway and Guy 1996).
See also Carmichael's Totient Function Conjecture, Totient Function
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 155, 1996.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 94, 1994.
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996.
Schlafly, A. and Wagon, S. ``Carmichael's Conjecture on the Euler Function is Valid Below
.''
Math. Comput. 63, 415-419, 1994.
Sloane, N. J. A. Sequence
A014573
in ``The On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html.
© 1996-9 Eric W. Weisstein