TOPICS

# Totient Valence Function

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).

Carmichael's Totient Function Conjecture, Sierpiński's Conjecture, Totient Function

## Explore with Wolfram|Alpha

More things to try:

## References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 155, 1996.Erdős, P. "Some Remarks on Euler's -Function." Acta Math. 4, 10-19, 1958.Ford, K. "The Distribution of Totients." Ramanujan J. 2, 67-151, 1998.Ford, K. "The Distribution of Totients, Electron. Res. Announc. Amer. Math. Soc. 4, 27-34, 1998.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. Sequences A002202/M0987, A007374/M1093, A014197, A014573, A058277, and A082695 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Totient Valence Function

## Cite this as:

Weisstein, Eric W. "Totient Valence Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TotientValenceFunction.html