Julia, 115 bytes

n->(a(n)=(n≥1&&(n==1?7:a(n-1)+gcd(n,a(n-1))));i=2;j=0;n>0&&while j<n x=a(i)-a(i-1);x>1&&(j+=1;println(x));i+=1end)

Ungolfed:

function a(n::Int)
    n ≥ 1 && (n == 1 ? 7 : a(n-1) + gcd(n, a(n-1)))
end

function f(n::Int)
    i = 2
    j = 0
    if n > 0
        while j < n
            x = a(i) - a(i-1)
            if x > 1
                j += 1
                println(x)
            end
            i += 1
        end
    end
end

Updated to allow n > 0 per the update to the spec, which added 5 bytes.

Julia, 110 bytes

n->(a(n)=(n≥1&&(n==1?7:a(n-1)+gcd(n,a(n-1))));i=2;j=0;while j<n x=a(i)-a(i-1);x>1&&(j+=1;println(x));i+=1end)

Ungolfed:

function a(n::Int)
    n ≥ 1 && (n == 1 ? 7 : a(n-1) + gcd(n, a(n-1)))
end

function f(n::Int)
    i = 2;
    j = 0;
    while j < n
        x = a(i) - a(i-1)
        if x > 1
            j += 1
            println(x)
        end
        i += 1
    end
end

Julia, 110 bytes

n->(a(n)=(n≥1&&(n==1?7:a(n-1)+gcd(n,a(n-1))));i=2;j=0;while j<n x=a(i)-a(i-1);x>1&&(j+=1;println(x));i+=1end)

Ungolfed:

function a(n::Int)
    n ≥ 1 && (n == 1 ? 7 : a(n-1) + gcd(n, a(n-1)))
end

function f(n::Int)
    i = 2;
    j = 0;
    while j < n
        x = a(i) - a(i-1)
        if x > 1
            j += 1
            println(x)
        end
        i += 1
    end
end

Julia, 115 bytes

n->(a(n)=(n≥1&&(n==1?7:a(n-1)+gcd(n,a(n-1))));i=2;j=0;n>0&&while j<n x=a(i)-a(i-1);x>1&&(j+=1;println(x));i+=1end)

Ungolfed:

function a(n::Int)
    n ≥ 1 && (n == 1 ? 7 : a(n-1) + gcd(n, a(n-1)))
end

function f(n::Int)
    i = 2
    j = 0
    if n > 0
        while j < n
            x = a(i) - a(i-1)
            if x > 1
                j += 1
                println(x)
            end
            i += 1
        end
    end
end

Updated to allow n > 0 per the update to the spec, which added 5 bytes.