totient valence function

Given an integer $n$, count how many integers $m$ in the set $\{n+1,n+2,\mathrm{\dots }{n}^2\}$ satisfy $\varphi (m)=n$. This is the totient valence of $n$, usually labelled $N_{\varphi }(n)$. (The only two special cases are 2 and 6, for which one has to look a little beyound $n^2$).

Robert Carmichael conjectured that $N_{\varphi }(n)=1$ never. Two sequences in Sloane’s OEIS that list numbers with higher totient valences than preceding numbers are A007374 and A097942.

Title	totient valence function
Canonical name	TotientValenceFunction
Date of creation	2013-03-22 15:50:57
Last modified on	2013-03-22 15:50:57
Owner	CompositeFan (12809)
Last modified by	CompositeFan (12809)
Numerical id	5
Author	CompositeFan (12809)
Entry type	Definition
Classification	msc 11A25