Totient valence

Euler's totient function (n) is defined as the number of positive integers not exceeding n that are relatively prime to n, where 1 is counted as being relatively prime to all numbers.  So, for example, (20) = 8 because the eight integers 1, 3, 7, 9, 11, 13, 17, and 19 are relatively prime to 20.  The table below shows values of (n) for n 20.

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
(n)	1	1	2	2	4	2	6	4	6	4	10	4	12	6	8	8	16	6	18	8

Euler's totient valence function v(n) is defined as the number of positive integers k such that (k) = n.  For instance, v(8) = 5 because only the five integers k = 15, 16, 20, 24, and 30 are such that (k) = 8.  The table below shows values of v(n) for n 16.  (For n not in the table, v(n) = 0.)

n	v(n)	k such that (k) = 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

Evaluate v(2¹⁰⁰⁰).