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It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of divisors of $n$. See also Mathematics Stack Exchange

Can we switch $\varphi(n)$ and $\sigma(n)$:

Is $n>9$ a prime if $$n\varphi(n)\equiv 2\pmod{\sigma(n)}?$$ (This congruence holds for $n=8$ and $9$.) I can only prove that this congruence does not hold for prime powers.

It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of divisors of $n$. See also Mathematics Stack Exchange

Can we switch $\varphi(n)$ and $\sigma(n)$:

Is $n>9$ a prime if $$n\varphi(n)\equiv 2\pmod{\sigma(n)}?$$ (This congruence holds for $n=8$ and $9$.) I can only prove that this congruence does not hold for prime powers.

It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of divisors of $n$.

Can we switch $\varphi(n)$ and $\sigma(n)$:

Is $n>9$ a prime if $$n\varphi(n)\equiv 2\pmod{\sigma(n)}?$$ (This congruence holds for $n=8$ and $9$.) I can only prove that this congruence does not hold for prime powers.

It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of divisors of $n$. See also Mathematics Stack Exchange

Can we switch $\varphi(n)$ and $\sigma(n)$:

Is $n>9$ a prime if $$n\varphi(n)\equiv 2\pmod{\sigma(n)}?$$ (This congruence holds for $n=8$ and $9$.) I can only prove that this congruence does not hold for prime powers.