{"schema":"vela.problem-packet.v0.1","problem":878,"statement":"If $n=\\prod_{1\\leq i\\leq t} p_i^{k_i}$ is the factorisation of $n$ into distinct primes then let\\[f(n)=\\sum p_i^{\\ell_i},\\]where $\\ell_i$ is chosen such that $n\\in [p_i^{\\ell_i},p_i^{\\ell_i+1})$. Furthermore, let\\[F(n)=\\max \\sum_{i} a_i\\]where the maximum is taken over all distinct $a_1,\\ldots,a_k\\leq n$ such that $(a_i,a_j)=1$ for $i\\neq j$ and all prime factors of each $a_i$ are prime factors of $n$.Is it true that, for almost all $n$,\\[f(n)=o(n\\log\\log n)\\]and\\[F(n) \\gg n\\log\\log n?\\]Is it true that\\[\\max_{n\\leq x}f(n)\\sim \\frac{x\\log x}{\\log\\log x}?\\]Is it true that (for all $x$, or perhaps just for all large $x$)\\[\\max_{n\\leq x}f(n)=\\max_{n\\leq x}F(n)?\\]Find an asymptotic formula for the number of $n<x$ such that $f(n)=F(n)$. Find an asymptotic formula for\\[H(x)=\\sum_{n<x}\\frac{f(n)}{n}.\\]Is it true that\\[H(x) \\ll x\\log\\log\\log\\log x?\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A339378","name":"Let n be a positive integer. For each prime divisor p of n, consider the highest power of p which does not exceed n. The sum a(n) of these powers is defined as the power-sum of n.","terms":"0,2,3,4,5,7,7,8,9,13,11,17,13,15,14,16,17,25,19,21,16,27,23,25,25,29,27,23,29,68,31,32,38,49,32,59,37,51,40,57,41,66,43,","url":"https://oeis.org/A339378"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}