N_ϕ(m) is the number of integers n for which the totient function ϕ(n) = m, also called the multiplicity of m. Erdős proved that if a multiplicity occurs once, it occurs infinitely often. The values of N_ϕ(m) for m = 1, 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 m = 1, 2, 4, 6, 8, 10, 12, 16, 18, 20, ... (OEIS A002202). The table below lists values for m<=50.

Carmichael's totient function conjecture | Sierpiński's conjecture | totient function