The user Avery Wenger said he came up with this problem randomly and I think it's very interesting!
Let $s:\mathbb N\to\mathbb N$ be the sum of digits function. Then, Wenger defines $w:\mathbb N\to\mathbb N$ given by $$w(n):=\min\{k\in\mathbb N: s^k(n)\;\mbox{has a single digit}\}\;\forall n\in\mathbb N$$
Now comes the really interesting part: the Wenger's Summation. $$W_m:=\sum_{n=1}^m (-1)^{n+1}w(n)$$
What is so interesting about it? Its amazingly weird graph!
As you can see, it behaves very poorly at first, looking kinda bumpy (jumps seem to occur on multiples of powers of ten). Soon, however, the graph starts to look very much like a line! Is this really Wenger's Summation asymptotic behavior? Is there a real constant $\omega$ (the Wenger's Constant) such that $$\lim_{m\to\infty} \frac{W_m}{m} = \omega\;?$$ If not, then what is Wenger's Summation asymptotic behavior? Can we at least show $(W_m)_m$ to be unbounded?
But of course, this is only the decimal Wenger's Summation, since we are taking number in decimal notation! What is so special about $10$, anyway? What about the binary Wenger's Summation? Does it behave just as oddly? It actually does worse!
So, for every possible basis $b$ we have a different Wenger's Summation $(W_m^b)_m$ and a different Wenger's Constant $\omega^b$. Ain't that wonderful? Is it true that, if $a<b$, then $\omega^a>\omega^b$? The alternating signals makes proving all this stuff so awkward.
I'm just amazed about this very innocent looking procedure generating such a beautiful cacophony. It all looks so useless and disconnected to... anything else. Do mathematics even has the tools to answer these questions? If you can find anything interesting about Wenger's Summations, please let me know it!
Attention: Be aware Avery Wenger did not call all these mathematical objects by his own name in the original post. I'm the only one responsible for that (Avery, if you are seeing this, thank you for sharing the formidable problem that randomly came to you).
Values for the Wenger's Summation were calculated through this rudimentary python script.
import matplotlib.pyplot as plt
import numpy as np
b = 4 # basis
def w(n):
ap = 0 # additive persistence
while n>=b:
aux = n
sum = 0
while aux>0:
sum+=(aux%b)
aux//=b
n = sum
ap+=1
return ap
S = 0 # Wenger's summation
W = []
for n in range(100000000):
S+=w(n)*(-1)**(n+1)
W.append(S)
Update: A comment pointed out the value $w(n)$ is called the additive persistence of $n$. The sequence $(w(n))_n$ (OEIS A031286 for base $10$) has some interesting properties. For example, every positive integer appears in it infinitely many times. Furthermore, the first occurrence of $N$ in it (OEIS A006050 for base $10$) is given by the recurrence formula $a(N)=2\cdot b^{\frac{a(N-1)-1}{b-1}}-1$, so its always odd, i.e., $N$ appears for the first time in the Wenger's Summation with a positive sign! This might be a first step in proving its boundlessness.
Also, inspired by the same comment, I decided to add the graph for the Quartenary Wenger's Summation, which twists our expectations by trending downwards (until it doesn't anymore)!
