Jelly, 20 19 18 bytes

S‘gṪ’ɗƇḟ¹Ṃṭ
1Ç¡>¹S

This is a full program.

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How it works

1Ç¡>¹S Main link. Argument: n (integer)

1 Set the return value to 1.
 Ç¡ Call the helper link n times.
 >¹ Compare the elements of the result with n.
 S Take the sum, counting elements larger than n.


S‘gṪ’ɗƇḟ¹Ṃṭ Helper link. Argument: A (array or 1)

S Take the sum of A.
 ‘ Increment; add 1.
 ɗƇ Drei comb; keep only elements k of [1, ..., sum(A)+1] for which the
 three links to the left return a truthy value.
 g Take the GCD of k and all elements of A.
 Ṫ Tail; extract the last GCD.
 ’ Decrement the result, mapping 1 to 0.
 ḟ¹ Filterfalse; remove the elements that occur in A.
 Ṃ Take the minimum.
 ṭ Tack; append the minimum to A.

Note that the generated sequence is $[1, mathbf0, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, dots]$. Since calling the helper link $n$ times generates a sequence of length $n + 1$, the $0$ is practically ignored.

Perl 6, 66 bytes

+grep *>$_,(1,2,first *gcd@_[*-1]>1,grep *∉@_,1..*...*)[^$_]

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Too slow on TIO for n = 1000.

JavaScript (ES6), 107 106 105 bytes

f=(n,a=[2,1],k=3)=>a[n-1]?0:a.indexOf(k)+(C=(a,b)=>b?C(b,a%b):a>1)(k,a[0])?f(n,a,k+1):(k>n)+f(n,[k,...a])

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How?

The helper function $C$ returns true if two given integers are not coprime:

C = (a, b) => b ? C(b, a % b) : a > 1

The array $a$ is initialized to $[2,1]$ and holds all values that were encountered so far in reverse order. Therefore, the last value is always $a[0]$.

To know if $k$ qualifies as the next term of the sequence, we test whether the following expression is equal to $0$:

a.indexOf(k) + C(k, a[0])

a.indexOf(k) is equal to either:

• 
  $-1$ if $k$ is not found in $a$
  


• 
  $0$ if $k$ is equal to the last value (in which case it's necessarily not coprime with it)


• some $ige 1$ otherwise

Therefore, a.indexOf(k) + C(k, a[0]) is equal to $0$ if and only if $k$ is not found in $a$ and $k$ is not coprime with the last value ($-1+true=0$).

Haskell, 89 82 bytes

Edit: -7 bytes thanks to @H.PWiz

f l=[n|n<-[3..],all(/=n)l,gcd(l!!0)n>1]!!0:l
g n=sum[1|x<-iterate f[2]!!(n-2),x>n]

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Husk, 16 bytes

#>¹↑¡§ḟȯ←⌋→`-Nḣ2

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Explanation

#>¹↑¡§ḟȯ←⌋→`-Nḣ2 Implicit input, say n=10
 ḣ2 Range to 2: [1,2]
 ¡ Construct an infinite list, adding new elements using this function:
 Argument is list of numbers found so far, say L=[1,2,4]
 N Natural numbers: [1,2,3,4,5,6,7...
 `- Remove elements of L: K=[3,5,6,7...
 ḟ Find first element of K that satisfies this:
 Argument is a number in K, say 6
 § → Last element of L: 4
 ⌋ GCD: 2
 ȯ← Decrement: 1
 Implicitly: is it nonzero? Yes, so 6 is good.
 Result is the EKG sequence: [1,2,4,6,3,9,12...
 ↑ Take the first n elements: [1,2,4,6,3,9,12,8,10,5]
# Count the number of those
 >¹ that are larger than n: 1

JavaScript, 93 91 bytes

Throws an overflow error for 0 or 1, outputs 0 for 2.

n=>(g=x=>n-i?a[++x]|(h=(y,z)=>z?h(z,y%z):y)(x,c)<2?g(x):(a[c=x]=++i,x>n)+g(2):0)(a=[i=c=2])

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MATL, 29 bytes

qq:2:w"GE:yX-y0)yZdqg)1)h]G>z

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Explanation:

	#implicit input, n, say 10
qq:	#push 1:8
2:	#push [1 2]. Stack: [1 .. 8], [1 2]
w	#swap top two elements on stack
"	#begin for loop (do the following n-2 times):
 GE:	#push 1...20. Stack: [1 2], [1..20]
 y	#copy from below. Stack:[1 2], [1..20], [1 2]
 X-	#set difference. Stack: [1 2], [3..20]
 y0)	#copy last element from below. Stack:[1 2], [3..20], 2
 yZd	#copy from below and elementwise GCD. Stack:[1 2], [3..20],[1,2,etc.]
 qg)	#select those with gcd greater than 1. Stack:[1 2], [4,6,etc.]
 1)	#take first. Stack:[1 2], 4
 h	#horizontally concatenate. Stack:[1 2 4]
 ]	#end of for loop
G>z	#count those greater than input
	#implicit output of result

Japt, 23 21 bytes

@_jX ªAøZ}f}gA=ì)Aè>U

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@_jX ªAøZ}f}gA=ì)Aè>U
 :Implicit input of integer U
 A :10
 ì :Digit array
 = :Reassign to A
@ }g :While the length of A < U+1, take the last element as X,
 :pass it through the following function & push the result to A
 _ }f : Find the first integer Z >= 0 that returns falsey
 jX : Is Z co-prime with X?
 ª : OR
 AøZ : Does A contain Z?
 ) :End loop
 Aè>U :Count the elements in A that are greater than U