Let g\,\ge \,2$g\,\ge \,2$. A real number is said to be g$g$-normal if its base g$g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let \varphi$\varphi$ denote Euler’s totient function, let \sigma$\sigma$ be the sum-of-divisors function, and let \lambda$\lambda$ be Carmichael’s lambda-function. We show that if f$f$ is any function formed by composing \varphi$\varphi$, \sigma$\sigma$, or \lambda$\lambda$, then the number

0.f\left( 1 \right)f\left( 2 \right)f\left( 3 \right)\,.\,.\,. $$0.f\left( 1 \right)f\left( 2 \right)f\left( 3 \right)\,.\,.\,.$$

obtained by concatenating the base g$g$ digits of successive f$f$-values is g$g$-normal. We also prove the same result if the inputs 1,2,3.... are replaced with the primes 2, 3, 5.... The proof is an adaptation of a method introduced by Copeland and Erdõs in 1946 to prove the 10-normality of 0:235711131719...

Let g\geqslant 2$g\geqslant 2$. A real number is said to be g$g$-normal if its base g$g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let \unicode[STIX]{x1D711}$\unicode[STIX]{x1D711}$ denote Euler’s totient function, let \unicode[STIX]{x1D70E}$\unicode[STIX]{x1D70E}$ be the sum-of-divisors function, and let \unicode[STIX]{x1D706}$\unicode[STIX]{x1D706}$ be Carmichael’s lambda-function. We show that if f$f$ is any function formed by composing \unicode[STIX]{x1D711}$\unicode[STIX]{x1D711}$, \unicode[STIX]{x1D70E}$\unicode[STIX]{x1D70E}$, or \unicode[STIX]{x1D706}$\unicode[STIX]{x1D706}$, then the number

\begin{eqnarray}0.f(1)f(2)f(3)\cdots\end{eqnarray} $$\begin{eqnarray}0.f(1)f(2)f(3)\cdots\end{eqnarray}$$

g$g$

f$f$

g$g$

1,2,3,\ldots$1,2,3,\ldots$

2,3,5,\ldots$2,3,5,\ldots$

0.235711131719\cdots \,$0.235711131719\cdots \,$