Totient Answers For The First 1000 Integers

The table below lists all totient (φ) answers to the first 1000 integers. We can be certain we're listing all since it is a mathematical fact that x/φ(x) is never more than the square root of x, (with only two exceptions, 2 and 6).

You might notice that in the table below, 1 is the only odd number in the n column. 1 is a special case. It is the only number that is its own φ value and the only odd number to ever be a φ value. So we need not list any other odd n, since the answer will always be the same: they're nontotients. There are even numbers that are never a φ value either. I don't know if there is a pattern to them. They are listed below as "Nontotients".

You might also notice that some numbers have more φ answers than others. A number that has more φ answers than any number before it is called a highly totient number. The first few highly totient numbers are 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, as you can verify from the table.

Values calculated with Mathematica 4, with the command Table[{n, Select[Range[n + 1, n^2 + 2], EulerPhi[#] == n &]}, {n, 2, 1000, 2}]. As noted above, 1 is a special case.