Powers of a Prime in the Image of Summatory Valuation Functions

J.-L. Nicolas asked how often a power of \(2\) belongs to the image of \(c(n) = \sum_{k\leq n} 2^{v_2(k)}\), where \(v_2\) is the \(2\)-adic valuation, and showed that the number \(A(x)\) of exponents \(a \leq x\) with \(2^a\) in the image and the number \(B(x)\) of the other exponents both tend to infinity. We prove 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 \((j2^j)_{j\geq 1}\). For every exponent there is at most one possible largest term \(K\), given by \(K + \lceil\log_2 K\rceil = a + 3\), and membership is then decided by an explicit recursion on pairs of integers at most \(a+3\), in at most \(a+1\) steps. The exponents without such a \(K\) and the exponents decided in at most one step form explicit infinite families, described completely, which contain the three families of Nicolas. The states of the recursion form a forest and the set of successful states is symmetric; this yields further explicit families and the bounds \(A(x) \geq (3.9655\ldots + o(1))\log_2 x\) and \(B(x) \geq (2.8178\ldots + o(1))\log_2 x\), whereas the families of Nicolas give the constants \(\frac12\) and \(\frac54\). The reduction, the unique candidate, the recursion and the symmetry extend to \(c_p(n) = \sum_{k\leq n} p^{v_p(k)}\) for every prime \(p\). Computations in exact integer arithmetic reproduce the table of Nicolas and give \(A(x)\) for \(x \leq 262000\), with \(A(x)/x\) between \(0.5132\) and \(0.8750\) for \(x = 1000, 2000, \ldots, 262000\).