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Assuming that $x$ is a real number, the function $f_n(x)$ is defined as follows: the value of $f_n(x)$ is equal to the number of bits before the first occurrence of $n$ consecutive zero bits in the bi …

Let $S$ denote an enumerated infinite collection of absolutely normal (i.e. normal in all integer bases greater than or equal to $2$) real numbers (with no repetitions) such that any natural number co …

Assuming that $i>0$, let $p(i)$ denote an $i$-th prime, i.e. $p(1)=2, p(2)=3, p(3) = 5$ etc. Let $$f(x, y)=\operatorname{floor}(x/2) \bmod 2^y,$$ i.e. $f(123, 4)=13, f(1234567, 8)=67, f(9876543210, 12 …

Assuming that $i \geq 0$, let $p_i$ denote an $i$-th prime: $p_0 = 2, p_1 = 3, \ldots$ Then $b_i$ denotes the second-to-last bit of $p_i$, i.e. $b_i = \left\lfloor p_i/2 \right\rfloor \bmod 2.$ The se …

Assuming that $n>0$, let $t_b(n)$ denote the base-$b$ representation of a natural number $n$, i.e. the tuple $$(d_k, d_{k-1}, \ldots, d_1, d_0)$$ such that $$n=d_kb^k+d_{k-1}b^{k-1}+\ldots+d_1b+d_0,$$ …

Let $X$ denote the sequence A200246: an $i$-th element of $X$ is equal to $w(p_i) \bmod 2$, where $w(p_i)$ is the number of ones in the base-$2$ representation of an $i$-th prime. The first $564163$ b …