1911 Encyclopædia Britannica/Number/Totients

26. Totients.&mdash;By the totient of $$n$$, which is denoted, after Euler, by $$\phi(n)$$, we mean the number of integers prime to $$n$$, and not exceeding $$n$$. If $$n=p^\alpha$$, the numbers not exceeding $$n$$ and not prime to it are $$p, 2p, \dots (p^\alpha-p), p^\alpha$$ of which the number is $$p^{\alpha-1}$$: hence $$\phi(p^\alpha)=p^\alpha-p^{\alpha-1}$$. If $$m, n$$ are prime to each other, $$\phi(mn) =\phi(m)\phi(n)$$; and hence for the general case, if $$n = p^\alpha q^\beta r^\gamma \dots,\,$$ $$\phi(n) = \textstyle \prod p^{\alpha-1}(p-1)$$, where the product applies to all the different prime factors of $$n$$. If $$d_1, d_2,\!\!\And\!\!\!\!\mathrm{c}.$$, are the different divisors of $$n$$,
 * $$\phi(d_1)+\phi(d_2)+\dots = n$$.

For example, $$15=\phi(15)+\phi(5)+\phi(3)+\phi(1)=8+4+2+1$$.