Japt, 21 bytes

It pains me to have to invoke a function without a shortcut - can't remember the last time I had to, if ever - but it worked out shorter than any other method I could come up with.

@W{WjY «ZøW ©ÓWaY}a}g

Try it

Husk, 22 bytes

!¡λḟ§&oε⌋o>1≠→L¹`-N);1

Try it online!

05AB1E, 26 bytes

2IŸεDU°2Ÿ¯KʒX¿}ʒXα1›}θDˆ}θ

Try it online or output the first $n$ terms as list. (NOTE: ° is obviously extremely slow, so replaced with T* in the TIO links ($10*n$ instead of $10^n$).)

Explanation:

2IŸ               # Create a list in the range [2, input]
   ε              # Map each value `y` to:
    DU            #  Store a copy of `y` in variable `X`
    °2Ÿ           #  Create a list in the range [10**`y`,2]
       ¯K         #  Remove all values already in the global_array
       ʒX¿}       #  Only keep values with a greatest common divider of 1 with `X`
       ʒXα1›}     #  Only keep values with an absolute difference larger than 1 with `X`
             θ    #  After these filters: keep the last (the smallest) element
              Dˆ  #  And add a copy of it to the global_array
   }θ             # After the map: only leave the last value
                  # (and output the result implicitly)

Japt, 33 bytes (non-competing?)^†

ò2 rÈc_aY >1«XøZ «Yk øZk}a+2}[] o

Try it online!

_{^† I fixed a bug in the Japt Interpreter while working on this solution. This meta post from a year ago deems this answer non-competing, but this newer meta post is pushing for more freedom in this. Regardless, I spent too much time on this to scrap it.}

Alice, 42 bytes

/oi
1@/2-&whq&[]w].q-H.t*n$K.qG?*h$KW.!kq

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Explanation

/oi
1@/.........

This is a standard template for programs that take a number as input, and output a number, modified to place a 1 on the stack below the input number.

The main part of the program places each number k in slot a(k) on the tape. The inner loop computes a(k), and the outer loop iterates over k until a(n) is computed.

1       push 1
i       take input
2-&w    repeat n-1 times (push return address n-2 times)
h       increment current value of k
q&[     return tape to position 0
]       move right to position 1
w       push return address to start inner loop
]       move to next tape position
.q-     subtract tape position from k
H       absolute value
.t*     multiply by this amount minus 1
n$K     if zero (i.e., |k-x| = 0 or 1), return to start of inner loop
.qG     GCD(k, x)
?       current value of tape at position: -1 if x is available, or something positive otherwise
*       multiply
h$K     if not -1, return to start of inner loop
W       pop return address without jumping (end inner loop)
.!      store k at position a(k)
k       end outer loop
q       get tape position, which is a(n)
o       output
@       terminate

VB.NET (.NET 4.5), 222 bytes

Function A(n)
Dim s=New List(Of Long)
For i=2To n
Dim c=2
While Math.Abs(i-c)<2Or g(c,i)>1Or s.Contains(c)
c+=1
End While
s.Add(c)
Next
Return s.Last
End Function
Function g(a, b)
Return If(b=0,a,g(b,a Mod b))
End Function

I had to roll your own GCD. I also couldn't figure out how to get it to not be an entire function.

GCD is always >= 1, so only need to ignore 1

Took out short-circuiting in the golf because it's shorter

Un-golfed

Function Answer(n As Integer) As Integer
    Dim seqeunce As New List(Of Integer)
    For i As Integer = 2 To n
        Dim counter As Integer = 2
        ' took out short-circuiting in the golf because it's shorter
        ' GCD is always >= 1, so only need to ignore 1
        While Math.Abs(i - counter) < 2 OrElse GCD(counter, i) > 1 OrElse seqeunce.Contains(counter)
            counter += 1
        End While
        seqeunce.Add(counter)
    Next
    Return seqeunce.Last
End Function

' roll your own GCD
Function GCD(a, b)
    Return If(b = 0, a, GCD(b, a Mod b))
End Function

Mathematica, 111 bytes

(s={};Table[m=2;While[m<3#,If[CoprimeQ[i,m]&&FreeQ[s,m]&&Abs[i-m]>1,s~AppendTo~m;m=3#];m++],{i,2,#}];s[[#-1]])&

Try it online! 2..23 (range mode)

Try it online! or 150 (distinct values)

Haskell, 61 bytes

f n=[i|i<-[2..],gcd i n<2,all((/=i).f)[2..n-1],abs(n-i)>1]!!0

Try it online!

I'm fairly new to Haskell, so any golfing tips are appeciated.

Thanks to Wheat Wizard for 2 bytes and nimi for 4 bytes

Explanation:

f n=[i|i<-[2..],gcd i n<2,all((/=i).f)[2..n-1],abs(n-i)>1]!!0
f n=                                                          -- define a function f :: Int -> Int
    [i|i<-[2..],                                              -- out of positive integers greater than two
                gcd i n<2,                                    -- gcd of i and n is 1
                          all((/=i).f)[2..n-1],               -- i isn't already in the sequence
                                               abs(n-i)>1]    -- and |n-i| > 1
                                                          !!0 -- take the first element