Concatenate N N Times

May 10, 2016

One solution is to treat the number as a string, replicate the characters then convert to a number:

(define (f n)
  (undigits (flatten (iterate n append (digits n)))))
> (f 12)

That looks simple but is actually fairly complicated, requiring several functions from the Standard Prelude.

Concatenated numbers form one of the Smarandache sequences, and are given by the formula \frac{n(10^{n D(n)}-1}{10^{D(n)}-1}, where D(n) is the number of digits in n:

(define (d n) ; number of digits in n
  (let loop ((n n) (k 0))
    (if (zero? n) k
      (loop (quotient n 10) (+ k 1)))))
(define (f n) ; concatenate n n times
    (* n (- (expt 10 (* n (d n))) 1))
    (- (expt 10 (d n)) 1)))
> (d 12)
> (f 12)

That’s both simpler and faster than the string solution. You can run the program at Besides the MathWorld page referenced above, this exercise is also discussed at A000461.

Pages: 1 2

10 Responses to “Concatenate N N Times”

  1. Tried this in perl – as perl has the built in operator… the concatenation is about 40% faster then the numeric method – and works for larger values of n….

  2. Simple iterative solution O(n):

    (defun self-cat-|O(n)| (n &optional (base 10))
      (let ((factor (expt base (ceiling (log n base)))))
        (loop :repeat n
              :for result := n :then (+ (* result factor) n)
              :finally (return result))))
    (list (self-cat-|O(n)| 7)
          (self-cat-|O(n)| 12)
          (self-cat-|O(n)| 1234))
    --> (7777777 121212121212121212121212 1234…1234)

    Notice that it performs n multiplications and n additions that quickly enough are bignum operations.

    This is typically an operation that can be reduced to O(logn), by noticing that when n is even:


    and when n is odd:


    Therefore we can write:

    (defun self-cat-|O(logn)| (n &optional (base 10))
      (labels ((cat (factor hi lo) (+ (* factor hi) lo))
               (rep (n nxn factor)
                   ((= 1 n)   nxn)
                   ((evenp n) (rep (/ n 2)
                                   (cat factor nxn nxn)
                                   (* factor factor)))
                   (t         (cat factor
                                   (rep (truncate  n 2)
                                        (cat factor nxn nxn)
                                        (* factor factor))
        (rep n n (expt base (ceiling (log n base))))))
    (list (self-cat-|O(n)| 7)
          (self-cat-|O(n)| 12)
          (self-cat-|O(n)| 35)
          (self-cat-|O(logn)| 7)
          (self-cat-|O(logn)| 12)
          (self-cat-|O(logn)| 35))
    --> (
  3. I doubt James tested his perl string version on non-single digit numbers; going thru strings is 6 times slower than the iterative arithmetic solution, and 1000 times slower than the dichotomy arithmetic solution:

    (defun self-cat-string (n)
      (read-from-string (format nil "~V@{~A~:*~}" n n)))
      (time (progn (self-cat-string 1234) nil))
      (time (progn (self-cat-|O(n)| 1234) nil))
      (time (progn (self-cat-|O(logn)| 1234) nil)))
    (progn (self-cat-string 1234) nil)
    took 12,088 microseconds (0.012088 seconds) to run.
          1,798 microseconds (0.001798 seconds, 14.87%) of which was spent in GC.
    During that period, and with 4 available CPU cores,
          9,303 microseconds (0.009303 seconds) were spent in user mode
          3,327 microseconds (0.003327 seconds) were spent in system mode
     10,542,560 bytes of memory allocated.
     33 minor page faults, 0 major page faults, 0 swaps.
    (progn (|SELF-CAT-O(n)| 1234) nil)
    took 3,842 microseconds (0.003842 seconds) to run.
           897 microseconds (0.000897 seconds, 23.35%) of which was spent in GC.
    During that period, and with 4 available CPU cores,
         2,745 microseconds (0.002745 seconds) were spent in user mode
         1,336 microseconds (0.001336 seconds) were spent in system mode
     2,582,768 bytes of memory allocated.
    (progn (|SELF-CAT-O(logn)| 1234) nil)
    took 77 microseconds (0.000077 seconds) to run.
    During that period, and with 4 available CPU cores,
         78 microseconds (0.000078 seconds) were spent in user mode
          3 microseconds (0.000003 seconds) were spent in system mode
     27,024 bytes of memory allocated.
  4. I meant the factors for 1234.

    Converting the number to a string is O(logn) divisions; concatenating the strings can be made O(n^2) memory accesses (the length of the string is n*n=n^2); reading the result is O(n^2) multiplications+additions. Total, the string version is O(n^2) multiplications, while the simple iterative is only O(n) multiplication and the dichotomic one is O(logn) multiplications.

  5. matthew said

    Here’s a couple of Python solutions, one short and snappy using strings, one using bignum arithmetic. The string solution is faster for smaller n, but with bignums winning for larger (>200) n – by n = 100000, bignums are 15 times faster, probably converting megabyte strings to ints doesn’t scale well.

    import sys
    import math
    import timeit
    def f1(n): return int(str(n)*n)
    def f2(n):
        k = 1 + int(math.log10(n))
        d = 10**k
        return (d**n-1)//(d-1)*n
    def test(f,n): return lambda: f(n)
    for i in range(1,1000): assert(f1(i) == f2(i))
    n = int(sys.argv[1])
    count = int(sys.argv[2])
  6. Daniel said

    Here’s a solution in Java that iteratively reads an additional character until the parsed integer times its string length equals the length of the entire string. I assumed inputs are valid (e.g., 333 is valid, 332 is not but my program would also return 3 for that). At return time, I could check for validity but didn’t bother.

    import java.math.BigInteger;
    public class Praxis {
        private static String parseN(String nn) {
            int len = nn.length();
            for (int i = 1; i <= len; i++) {
                String str = nn.substring(0, i);
                BigInteger n = new BigInteger(str);
                int chars = n.multiply(BigInteger.valueOf(str.length())).intValue();
                // assumes input is valid.
                // could alternatively check in condition below.
                if (chars == len) {
                    return n.toString();
                } else if (chars > len) {
                    return ""; // error
            return ""; // error
        public static void main(String[] args) {


  7. Daniel said
    public static void main(String[] args) {


  8. Daniel said

    > “Your task is to write a program that calculates the number that is concatenated to itself the number of times as the number is (that’s hard to say).”

    I just looked at the solution and skimmed some other solutions. I think I interpreted the task slightly differently than others. It seems like the typical interpretation is that the input is N, and the output is N concatenated N times. I thought the input was N concatenated N times, and the output was N. I re-read the task and think it can be interpreted either way (its ambiguous, like many natural language sentences).

  9. Chris Judge said

    Building the output in an array of chars is wicked fast. The following – in c# – completes in 1 millisecond with n = 100000:

            static string viaDoubling (string input) {
                int n = int.Parse(input);
                long blockLen = input.Length;
                long outLen = n * blockLen;
                char[] output = new char[outLen];
                input.CopyTo(0, output, 0, input.Length);
                while (blockLen < outLen / 2) {
                    Array.Copy(output, 0, output, blockLen, blockLen);
                    blockLen *= 2;
                Array.Copy(output, 0, output, blockLen, outLen - blockLen);
                return new string(output);

    A math solution requires approximately 40000 milliseconds for the same n = 100000:

            static string viaMath(string input) {
                int n = int.Parse(input);
                int m = (int) Math.Pow(10, input.Length);
                BigInteger s = 0;
                for (int i = 0; i < n; i++) {
                    s *= m;
                    s += n;
                return s.ToString();
  10. Sinan said

    I have two solutions both using python and does substring comparison although numeric comparisons are cool.

    First version is kind of greedy approach

    def find_repetitions(s):
        for i in range(1, len(s) / 2 + 1):
            frag = s[:i]
            frags = s.split(frag)
            if not any(frags):
                return frag, len(frags) - 1

    Second version uses indexes to compare substrings – it resembles to dynamic programming solutions.

    def find_repetitions2(s):
        x = len(s[:len(s) / 2 + 1])
        y = len(s)
        for i in range(1, x):
            for j in range(i, y, i):
                if i > y - j:
                for k in range(i):
                    if s[k] != s[j + k]:
                return s[:i], y / i

    Performance comparison, measured by ipython’s timeit builtin function.

    First version seems to do better at first glance

    In [_]: %timeit find_repetitions('xcv' * 1024)
    10000 loops, best of 3: 113 µs per loop
    In [_]: %timeit find_repetitions(string.letters * 1024)
    100 loops, best of 3: 6.22 ms per loop
    In [_]: %timeit find_repetitions2('xcv' * 1024)
    1000 loops, best of 3: 599 µs per loop
    In [_]: %timeit find_repetitions2(string.letters * 1024)
    100 loops, best of 3: 7.19 ms per loop

    But i suspected that second version would do better when repeated substring becomes much larger, and it indeed did perform better

    In [_]: %timeit find_repetitions(''.join(random.choice(string.letters) for _ in range(500)) * 1024)
    1 loops, best of 3: 387 ms per loop
    In [_]: %timeit find_repetitions2(''.join(random.choice(string.letters) for _ in range(500)) * 1024)
    10 loops, best of 3: 87.9 ms per loop

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