## Factoids #1

#### IV) commutative, non-associative operations

For any natural number n, let $\mathrm{Op}_n:=\left\{\star:\mathbb{Z}^2_n\to\mathbb{Z}_n\right\}$ denote the set of all operations on a set of that order. An operation  shall be called commutative iff $\mathrm{commut}(\star):\Leftrightarrow\forall\,x,y\in\mathbb{Z}_n:x\star y=y\star x$ and be called associative iff $\mathrm{assoc}(\star):\Leftrightarrow\forall\,x,y,z\in\mathbb{Z}_n:x\star(y\star z)=(x\star y)\star z$ holds.

With the above defined, one may study $\mathrm{CnA}_n:=\{\star\in\mathrm{Op}_n:\mathrm{commut}(\star)\land\lnot \mathrm{assoc}(\star)\}$. For n = 2, this set is nonempty for the first time, containing a manageable two elements, by name

$\mathrm{CnA}_2=\Big\{\mathrm{nor}:(x,y)\mapsto 1+xy, \quad\mathrm{nand}:(x,y)\mapsto (1+x)\cdot(1+y)\Big\}$.

However, based on the superexponential nature of $\#\mathrm{Op}_n=\#\mathbb{Z}_n^{\mathbb{Z}_n^2}=\#\mathbb{Z}_n^{{\#\mathbb{Z}_n}^2}=n^{n^2}$, the sequence $\mathrm{A079195}_n:=\#\mathrm{CnA}_n$ likely also grows rather quickly, OEIS only listing four members;

$\mathrm{A079195}=(0, 2, 666, 1\,047\,436, \dots)$.

Based on this limited numerical evidence, I would suspect the commutative yet non-associative operations to be rather sparse, i.e.

$\lim\limits_{n\to\infty}\mathrm{A079195}_n\cdot\left(\#\mathrm{Op_n}\right)^{-1}=0\mod\square$.

Analysis source: operations.hs

(Non-)commutative and (non-)associative operations have also been studied nearly twenty years ago by Christian van den Bosch, author of OEIS sequence A079195. Unfortunately, their site appears to be down, which is where they hosted Closed binary operations on small sets (resource found on web.archive.org).

#### V) Arbitrary polynomial extremum difference

Let ε > 0 be an arbitrary distance, define $g := -4x^4-x^3+8x^2+3x-4$. Then $f:=\epsilon\cdot 4^{-1}\cdot g$ has two local maxima at -1 and 1, whose vertical distance is ε.

#### VI) Digit sum roots

It holds that $\mathrm{ds}_{10}(108^{12})=108$.

## Extending A056154

Five weeks of work including over six days of dedicated number crunching come to fruition as the thirteenth member of OEIS sequence A056154 is published,

$\mathrm{A056154}(13) = 49\,094\,174.$

Sequence A056154 is defined as binary exponents which have a ternary representation invariant under endomorphic addition modulo permutation, more formally

\begin{aligned} a \in \mathrm{A056154}\,:\Longleftrightarrow\, &a,\log_2(a)\in\mathbb{N}\,\land\,\exists\,\sigma\in\mathrm{Sym}(\{0,\dots,\lfloor\log_3(a+a)\rfloor\}):\\ &\forall\,j\in\mathrm{dom}\,\sigma:\Big\lfloor (a+a)\cdot 3^{-j} \Big\rfloor \equiv \Big\lfloor a\cdot 3^{-\sigma(j)} \Big\rfloor \mod 3. \end{aligned}

Due to the exponentially defined property, testing a given $p\in\mathbb{N}$ for membership quickly becomes non-trivial, as the trits of $2^p$ enter the billions.
As an example, $2^{49\,094\,174}$ requires 30’974’976 trits. Assuming three thousand trits per page and two hundred pages per book, a ternary print-out of said number would require fifty-two books, filling a few book shelves.

For a discussion of the methodology I used to perform the search which lead to the discovery of $\mathrm{A056154}(13)$, I refer to my paper Extending A056154.

## A325902

Fifty is a peculiar integer.
When looking at its neighbors — the largest integer strictly beneath and the smallest strictly above –, more specifically their prime factorization, one finds

$49=\underbrace{7^2<50<3\cdot 17}_{7+7+3=17}=51,$

notably there exists a partition of the neighbor’s factors into two multisets such that both parts’ sums equal another.

Positive integers with the above described property can be found in my most recent addition to the OEIS: sequence A325902.