{"schema":"vela.problem-packet.v0.1","problem":315,"statement":"Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $\\sum_{k\\geq 1}\\frac{1}{u_k+1}$ and $u_k=\\lfloor c_0^{2^k}+1\\rfloor$ for $k\\geq 1$, where\\[c_0=\\lim u_n^{1/2^n}=1.264085\\cdots.\\]Let $a_1<a_2<\\cdots $ be any other sequence with $\\sum \\frac{1}{a_k}=1$. Is it true that\\[\\liminf a_n^{1/2^n}<c_0=1.264085\\cdots?\\]","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A000058","name":"Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2.","terms":"2,3,7,43,1807,3263443,10650056950807,113423713055421844361000443,12864938683278671740537145998360961546653259485195807","url":"https://oeis.org/A000058"},{"id":"A076393","name":"Decimal expansion of Vardi constant arising in the Sylvester sequence.","terms":"1,2,6,4,0,8,4,7,3,5,3,0,5,3,0,1,1,1,3,0,7,9,5,9,9,5,8,4,1,6,4,6,6,9,4,9,1,1,1,4,5,6,0,1,7,9,2,0,9,0,6,5,5,3,3,1,5,3,4,5,","url":"https://oeis.org/A076393"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}