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

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" | $60$
|}
{{OEIS|A001172}}

Proof:
{{begin-eqn}}
{{eqn | l = 6
      | r = 3 + 3
      | c = $1$ way
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 8
      | r = 5 + 3
      | c = $1$ way
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 10
      | r = 7 + 3
      | c = $2$ ways
}}
{{eqn | r = 5 + 5
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 12
      | r = 7 + 5
      | c = $1$ way
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 14
      | r = 11 + 3
      | c = $2$ ways
}}
{{eqn | r = 7 + 7
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 16
      | r = 13 + 3
      | c = $2$ ways
}}
{{eqn | r = 11 + 5
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 18
      | r = 13 + 5
      | c = $2$ ways
}}
{{eqn | r = 11 + 7
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 20
      | r = 17 + 3
      | c = $2$ ways
}}
{{eqn | r = 13 + 7
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 22
      | r = 19 + 3
      | c = $3$ ways
}}
{{eqn | r = 17 + 5
      | c = 
}}
{{eqn | r = 11 + 11
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 24
      | r = 19 + 5
      | c = $2$ ways
}}
{{eqn | r = 13 + 11
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 26
      | r = 23 + 3
      | c = $3$ ways
}}
{{eqn | r = 19 + 7
      | c = 
}}
{{eqn | r = 13 + 13
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 28
      | r = 23 + 5
      | c = $2$ ways
}}
{{eqn | r = 17 + 11
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 30
      | r = 23 + 7
      | c = $3$ ways
}}
{{eqn | r = 19 + 11
      | c = 
}}
{{eqn | r = 17 + 13
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 32
      | r = 29 + 3
      | c = $2$ ways
}}
{{eqn | r = 19 + 13
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 34
      | r = 31 + 3
      | c = $4$ ways
}}
{{eqn | r = 29 + 5
      | c = 
}}
{{eqn | r = 23 + 11
      | c = 
}}
{{eqn | r = 17 + 17
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 36
      | r = 31 + 5
      | c = $4$ ways
}}
{{eqn | r = 29 + 7
      | c = 
}}
{{eqn | r = 23 + 13
      | c = 
}}
{{eqn | r = 19 + 17
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 38
      | r = 31 + 7
      | c = $2$ ways
}}
{{eqn | r = 19 + 19
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 40
      | r = 37 + 3
      | c = $3$ ways
}}
{{eqn | r = 29 + 11
      | c = 
}}
{{eqn | r = 23 + 17
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 42
      | r = 37 + 5
      | c = $4$ ways
}}
{{eqn | r = 31 + 11
      | c = 
}}
{{eqn | r = 29 + 13
      | c = 
}}
{{eqn | r = 23 + 19
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 44
      | r = 41 + 3
      | c = $3$ ways
}}
{{eqn | r = 37 + 7
      | c = 
}}
{{eqn | r = 31 + 13
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 46
      | r = 43 + 3
      | c = $4$ ways
}}
{{eqn | r = 41 + 5
      | c = 
}}
{{eqn | r = 29 + 17
      | c = 
}}
{{eqn | r = 23 + 23
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 48
      | r = 43 + 5
      | c = $5$ ways
}}
{{eqn | r = 41 + 7
      | c = 
}}
{{eqn | r = 37 + 11
      | c = 
}}
{{eqn | r = 31 + 17
      | c = 
}}
{{eqn | r = 29 + 19
      | c = 
}}
{{end-eqn}}
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 $60$.
{{qed}}