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.