Sum of digits iterated - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:10:59Z http://mathoverflow.net/feeds/question/27010 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27010/sum-of-digits-iterated Sum of digits iterated Wadim Zudilin 2010-06-04T04:36:31Z 2010-06-04T05:58:45Z <p><strong>Original version.</strong> I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of digits $n_2=S(n)=S_{10}(n)$ in the decimal system. If the newer number $n_2$ is greater than $10$, then compute the sum $n_3=S(n_2)$ of its digits, and continue this iteration $n_k=S(n_{k-1})$ unless you get a number $n^* =n_\infty$ in the range $1\le n^* \le 9$. Is $n^*$ uniformly distributed in the set $\lbrace 1,2,\dots,9\rbrace$? If this is not true in the decimal systems, what can be said in the other systems?</p> <p>I just learned yesterday about the Feng shui system of determining what kind of problems/advantages one can get according to the house number, say $n$, of his/her home. This depends on the above $n^*$. I do not seriously count on the conclusions but I am curious whether $n^*$ is sufficiently democratic.</p> <p><strong>Edit.</strong> The question was immediately realized as obvious, because $n^*$ is the residue modulo $9$ (with 0 replaced by 9), and this works in any base as well. So the Feng shui function is really trivial, but one can deal with less trivial ones.</p> <p>Let me fix $m$ and define $Q_m(n)$ as the sum of $m$th powers of decimal digits of a positive integer $n$. What can be said about the sequence of iterations $n_k=Q_m(n_{k-1})$ for a given integer $n_0$? How long can the (minimal) period be for a fixed $m$? And what can be said about the distribution of the purely periodic tails?</p> <p>I hope that the question is still elementary.</p> http://mathoverflow.net/questions/27010/sum-of-digits-iterated/27017#27017 Answer by Gerry Myerson for Sum of digits iterated Gerry Myerson 2010-06-04T05:52:01Z 2010-06-04T05:52:01Z <p>A starting place might be <a href="http://www.research.att.com/~njas/sequences/A005188" rel="nofollow">http://www.research.att.com/~njas/sequences/A005188</a> which lists $n$-digit numbers $r$ with $Q_n(r)=r$, and has references to related oddities. </p> http://mathoverflow.net/questions/27010/sum-of-digits-iterated/27018#27018 Answer by Nurdin Takenov for Sum of digits iterated Nurdin Takenov 2010-06-04T05:58:45Z 2010-06-04T05:58:45Z <p>The case $m=2$ appears in Hugo Steinhaus's "One Hundred Problems In Elementary Mathematics", problem 2(at least in Russian edition of 1986). Either sequence will come to 1 and stay here, or will enter to the cycle (145,42,20,4,16,37,58,89) </p>