Form of odd primes with totients co-prime to $5$?

Question

I'm trying to generate odd prime numbers such that their totients are co-prime with $5$. I want to know if what I'm doing is correct, or can be proven.

1. Assume $2, 3$ and $5$ are primes
2. Check if $10i - 3, 10i - 1, 10i + 3$ for $i$ greater or equal to $1$

now I did the following to get those forms and I wonder if the approach is correct:

1. Assume every integer is in the form of $5n-2, 5n-1, 5n, 5n+1, 5n+2$

2. substitute each of them with $k$ in $2k+1,$ and get $10n-3, 10n-1, 10n+1, 10n+3, 10n+5$

3. remove the forms that don't form prime for n in positive integers and have the forms reduced to $10n-3, 10n-1, 10n+1, 10n+3$

4. totient of a prime $p$ is $p-1,$ so subtract $1$ from each of the forms and see if they're divisible by $5$ for every positive integer $n, 10n-4, 10n-2, 10n+2$.

5. check numbers of the form $10i-3, 10i-1$ and $10i+3$ for primality.