  Motivated by a question of Venkataramana, we consider the greatest common
divisor of $\phi(f(n))$ where $f$ is a primitive polynomial with integer
coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's
hypothesis, we establish that this gcd may be bounded just in terms of the
degree of the polynomial $f$. Unconditionally we establish such a bound for
quadratic polynomials, as well as polynomials that split completely into linear
factors. The paper also addresses a question of Calegari, and establishes that
there are infinitely many $n$ such that $n^2+1$ is not divisible by any prime
$\equiv 1 \bmod 2^m$ provided $m$ is a large fixed integer.
