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.

Digit sums

Interessant war es auch, drei aufeinanderfolgende Zahlen zu nehmen, von denen die größte durch drei teilbar sein musste, sie zu addieren und aus dem Ergebnis so lange die Quersumme zu bilden, bis eine einstellige Zahl übrig blieb. Diese Zahl war immer sechs.
— Child, Lee: Der Anhalter. München: Blanvalet, 2015; p. 73.

Jack Reacher’s at most tangentially to interpreting the sergeant’s reply related base ten factoid’s formal form is

\forall n\in\mathbb{N}^+:\mathrm{fds}_{10}\left(\sum\limits_{j=0}^2 3n-j\right) = 6,

where \mathrm{fds}_{10} represents the final digit sum in base ten.

A proof of the above claim together with the underlying digit sum results is presented in digit_sums.pdf (source: digit_sums.tex).