Jonathan. Frech’s WebBlog

Extending A056154 (#221)

Jonathan Frech

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,

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

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

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

Due to the exponentially defined property, testing a given $p\in\mathbb{N}$$p\in\mathbb{N}$ for membership quickly becomes non-trivial, as the trits of $2^p$$2^p$ enter the billions.
As an example, $2^{49\,094\,174}$$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)$$\mathrm{A056154}(13)$, I refer to my paper Extending A056154.