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Totient Valence Function

$N_\phi(m)$ is the number of Integers $n$ for which $\phi(n)=m$, also called the Multiplicity of $m$ (Guy 1994). The table below lists values for $\phi(N)\leq 50$.

$\phi(N)$ multiplicity $N$
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 $\phi(N)$ with multiplicities up to 100 follows (Sloane's A014573).

$M$ $\phi$ $M$ $\phi$ $M$ $\phi$ $M$ $\phi$
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 $N_\phi(m)\geq 2$ (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 $n$, there exists $m\not=n$ such that $\phi(n)=\phi(m)$ (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 $10^{10,000,000}$.'' 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.



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© 1996-9 Eric W. Weisstein
1999-05-26