J(7,1) = 7      // people are counted out in order 1 2 3 4 5 6 [7]
J(7,2) = 7      // people are counted out in order 2 4 6 1 5 3 [7]
J(7,3) = 4      // see above
J(7,11) = 1
J(77,8) = 1
J(123,12) = 21

{{@+)%}+,*)}:f;

{          # Function declaration
           # Stack: n k
  {        # Stack: g(i-1,k) i-1 k
    @+)%  # Stack: g(i,k)
  }+       # Add, giving stack: n {k block}
  ,*      # Fold {k block} over [0 1 ... n-1]
  )        # Increment to move from 0-based to 1-based indexing
}:f;

J=lambda n,k:n<2or(J(n-1,k)+k-1)%n+1

>>> J(7,3)
4
>>> J(77,8)
1
>>> J(123,12)
21

1~f~_=1
n_~f~k_:=Mod[f[n-1,k]+k,n,1]

j(n,k){return n>1?(j(n-1,k)+k-1)%n+1:1;}

j(n,k,c,i,r){char o[999];memset(o,1,n);for(c=k,i=0;r;++i)(c-=o[i%=n])||(o[i]=!r--,c=k);
return i;}

[d1-d1<glk+r%1+]dsg?1-skrxp

# comment shows what is on the stack and any other effect the instructions have
[   # n
d   # n, n
1-  # n-1, n
d   # n-1, n-1, n
1<g # g(n-1), n ; g is executed only if n > 1, conveniently g(1) = 1
lk+ # g(n-1)+(k-1), n; remember, k register holds k-1
r%  # g(n-1)+(k-1) mod n
1+  # (g(n-1)+(k-1) mod n)+1
]
dsg # code for g; code also stored in g
?   # read user input => k, n, code for g
1-  # k-1, n, code for g
sk  # n, code for g; k-1 stored in register k
r   # code for g, n
x   # g(n)
p   # prints g(n)

j=.[:{.@{:]([:}.]|.~<:@[|~#@])^:(<:@#)>:@i.@[

j=.1:`(1+[|1-~]+<:@[$:])@.(1<[)

7 j 3
4
   123 j 12
21

:k;0:^;(,{))^k+%:^;}/^)

{1$1>{1$(1$^1$(+2$%)}1if@@;;}:^;

def j(n,k){n>1?(j(n-1,k)+k-1)%n+1:1}

j n=q$cycle[0..n]
q l@(i:h:_)k|h/=i=q(drop(k-1)$filter(/=i)l)k|1>0=i

J=function(n,k)ifelse(n<2,1,(J(n-1,k)+k-1)%%n+1)

def?(n:Int,k:Int):Int=if(n<2)1 else(?(n-1,k)+k-1)%n+1

j(n,k){
    int i=0,c=k,r=n,*p=calloc(n,8);
    for(;p[i=i%n+1]||--c?1:--r?p[i]=c=k:0;);
    return i;
}

1&|.&.#:

       1&|.&.#: 10
    5

       1&|.&.#: 69
    11

        1&|.&.#: 123456
    115841

        1&|.&.#: 123245678901234567890x NB. x keeps input integral
    98917405212792722853

All credit to Roger Hui, co-inventor of J and all-round uber-genius
www.jsoftware.com for free j software across many platforms

Explanation
    (J works right-to-left)
     #:       convert input to binary
     &.       a&.b <=> perform b, perform a, perform reverse of b
     1&|.     rotate bitwise one bit left

So
    1&|.&.#: 10

    a. #:            convert input (10) TO binary -> 1 0 1 0
    b. 1&|.          rotate result 1 bit left -> 0 1 0 1
    c. due to the &. perform convert FROM binary -> 5 (answer)

f:{$[x=1;1;1+mod[;x]f[x-1;y]+y-1]}

q)f .'(7 1;7 2;7 3;7 11;77 8;123 12)
7 7 4 1 1 21

J=->n,k{n<2?1:(J(n-1,k)+k-1)%n+1}

1#_=1
n#k=mod((n-1)#k+k-1)n+1

function f(a,b){return a<2?1:(f(a-1,b)+b-1)%a+1}

f=(a,b)=>a<2?1:(f(a-1,b)+b-1)%a+1

L[Dg#²<FÀ}¦

L           # Range 1 .. n
 [Dg#       # Until the array has a length of 1:
     ²<F }  #   k - 1 times do:
        À   #     Rotate the array
          ¦ #   remove the first element