# Repeated application of narcissistic function

Over on Code Golf.SE, I've asked this challenge about the looping behaviour of a function related to narcissistic numbers. In the question, I define the function as, for a natural number $$x = d_1d_2...d_n$$ where $$d_i$$ is a single digit between $$0$$ and $$9$$ and $$x$$ has $$n$$ digits:

$$f(x) = \sum_{i=1}^n d_i^n$$

In this case, a number is narcissistic if it is a fixed point of $$f(x)$$ i.e. $$f(x) = x$$. However, the question is about repeated application of this function. For example, repeatedly applying $$f$$ to $$x = 127$$ leads to the following chain of numbers:

$$127,352,160,217,352,160,217,...$$

and the triplet $$352, 160, 217$$ repeats ad infinitum from here. I have two hypotheses which I believe to be correct but have no idea how to prove:

1. For all natural numbers $$x > 0$$, repeated application of $$f$$ eventually leads to a repeating loop (counting a fixed point as a "loop" of 1 element).
2. The length of this loop is always less than or equal to $$14$$. For example, $$x = 147$$ is the lowest number that yields $$14$$, with a loop of

$$537059, 681069, 886898, 1626673, 1665667, 2021413, 18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340$$

My brute forcing attempts have shown that this is true for all $$x < 10^5$$, and is currently (at time or writing) at around $$x = 64000$$ without having found a counter example to either hypothesis.

This is a program which takes an input $$x$$ and outputs the repeated application of $$f$$ to $$x$$ until a repeated value is found, the loop of $$x$$ and the length of said loop, if you'd like to test out some other numbers.

• Since $f(x) \leqslant n\cdot 9^n$ if $x$ has $n$ digits, and the latter means $10^{n-1} \leqslant x < 10^{n}$, we have $f(x) < x$ whenever $\frac{\log n + \log 10}{n} < \log \frac{10}{9}$, which is the case for $n \geqslant 61$. Point 1 follows, and the smallest number in each loop has at most $60$ digits. One can reduce the bound a bit, and since one needs only consider nondecreasing digit sequences for starting values, the search space is further reduced. But it's still so large that a search for the longest loop(s) won't be quick. Nov 6, 2020 at 18:53
• @DanielFischer A quick check also shows that if $x$ has precisely $60$ digits, then $f^k(x)$ has fewer than $60$ digits for some $k\leq4$, and so to find loops it suffices to look for $x$ such that $f(x)\geq x$ where $x$ has at most $59$ digits. Sorting the digits as you suggest leaves $$\tbinom{59+10}{10}=340032449328\approx3.40\cdot10^{11},$$ starting values to check. I'm not much of a programmer, so I don't know how feasible this is. Nov 8, 2020 at 21:40

After checking (hopefully) all possibilities, for number bases 3 to 10, I have found the following loops:

Base-10: https://pastebin.com/8extqeGy

Count of narcissistic loops in
different number bases:

Base
3  4  5  6  7  8  9 10
+ ------------------------
1 |   6 12 18 31 60 63 59 89
2 |   -  -  2  6 17 11 14 16
3 |   -  -  -  -  7  4 11  7
4 |   -  -  1  2  6  5  8  5
5 |   -  -  -  1  -  1  5  4
6 |   -  -  1  1  3  2  1  5
7 |   -  -  -  -  2  2  2  3
8 |   -  -  -  -  2  -  1  -
9 |   -  -  -  -  2  -  -  2
10 |   -  -  -  -  -  1  -  2
11 |   -  -  -  -  -  -  -  -
12 |   -  -  -  -  -  -  1  2
13 |   -  -  -  -  -  1  -  1
14 |   -  -  -  -  -  -  -  2
15 |   -  -  -  -  -  1  -  -
16 |   -  -  -  -  -  -  1  -
17 |   -  -  -  -  -  -  -  -
18 |   -  -  -  -  -  -  -  1
19 |   -  -  -  -  -  -  -  -
...|
31 |   -  -  -  -  1  -  -  -
^ Loop length


The largest singleton is in base-10:

$$115132219018763992565095597973971522401=1^{39}+1^{39}+5^{39}+1^{39}+3^{39}+2^{39}+2^{39}+1^{39}+9^{39}+0^{39}+1^{39}+8^{39}+7^{39}+6^{39}+3^{39}+9^{39}+9^{39}+2^{39}+5^{39}+6^{39}+5^{39}+0^{39}+9^{39}+5^{39}+5^{39}+9^{39}+7^{39}+9^{39}+7^{39}+3^{39}+9^{39}+7^{39}+1^{39}+5^{39}+2^{39}+2^{39}+4^{39}+0^{39}+1^{39}$$

Here's the loop of 18 in base-10:

10074069541108119620821,18935428129475061827932,28415252997720554102092,
27233895488449729771663,28470768372157693427350,10181976920394277400584,
27852314084047558510219,10688610940325001073897,19553092616993382187366,
36662895609794210663524,27211241280778321507795,10180397389391156436853,
28388673910517178592951,29032929335610790615729,44371031395529147765128,
18399103996082548276483,37841895989779904859691,64484670033935168320603


Here's the loop of 31 in base-7:

105343,115515,253446,414526,412436,321136,260316,541235,210236,255345,
303626,543325,212326,255466,1044166,5015551,2441041,263452,400006,300652,
351614,366436,1144156,3215311,462622,554356,556516,1122466,4620541,3215551,
2004031

• This is another number with loop $18$, which is definitely interesting Nov 10, 2020 at 22:17