{"id":"hexagon/2611.00027","pdf":"https://hexagonmath.org/pdf/2611.00027v1","url":"https://hexagonmath.org/abs/2611.00027v1","tags":["math.NA","msc:11A63","msc:11A25","msc:11B83","msc:11N37","$2$-adic valuation","Arithmetic functions","Binary expansion","Open questions of Nicolas","Polynomials over $\\mathbb{F}_2$"],"score":0,"title":"Powers of a Prime in the Image of Summatory Valuation Functions","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Frank Vega"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-03","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}