Extending A056154

2019-11-02, post № 221

mathematics, #discovery, #OEIS, #paper, #ternary

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.