Powers of a Prime in the Image of Summatory Valuation Functions
Source: hexagonmath.org/abs/2611.00027v1
J.-L. Nicolas asked how often a power of $2$ belongs to the image of $c(n)=\sum_{k\le n}2^{v_2(k)}$, where $v_2$ is the $2$-adic valuation, and showed that both the number $A(x)$ of exponents $a\le x$ with $2^a$ in the image and the number $B(x)$ of the other exponents tend to infinity. We prove the identity $c(n)=n+\frac12\sum_{i}i\,2^{i}$, the sum running over the positions of the binary digits $1$ of $n$, so that $2^a$ lies in the image of $c$ if and only if $2^{a+3}$ is a sum of distinct terms of the superincreasing sequence $(j2^j)_{j\ge1}$. For every exponent there is at most one…
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