Smallest Positive Integer which is Sum of 2 Odd Primes in n Ways

Theorem
The sequence of positive integers $n$ which are the smallest such that they are the sum of $2$ odd primes in $k$ different ways begins as follows:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $k$ ! align="right" style = "padding: 2px 10px" | $n$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $22$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $34$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $48$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $60$
 * }
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $48$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $60$
 * }
 * }

Proof
From Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways, the smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$.