{"problem":{"name":"A2. Product of Triples (Hard Version)","description":{"content":"Many great mathematicians have sequences named after them. Timmy is a great mathematician, so he created a sequence called $t$, but he needs help to compute its values. Let $t_i$ be the number of unor","description_type":"Markdown"},"platform":"Codeforces","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"CF10264A2"},"statements":[{"statement_type":"Markdown","content":"Many great mathematicians have sequences named after them. Timmy is a great mathematician, so he created a sequence called $t$, but he needs help to compute its values. Let $t_i$ be the number of unordered triples $(a, b, c)$ where $a <= b <= c$ and $a dot.op b dot.op c = i$. For all $i$ from $1$ to $n$, find and print $t_i$.\n\nThe only line contains a single integer $n$ $(1 <= n <= 10^4)$.\n\nFor all $i$ from $1$ to $n$, print $t_i$.\n\nThere are $3$ triples that have product $8$: $(1, 1, 8)$, $(1, 2, 4)$, and $(2, 2, 2)$. However, there is only $1$ triple that has product $7$: $(1, 1, 7)$.\n\n## Input\n\nThe only line contains a single integer $n$ $(1 <= n <= 10^4)$.\n\n## Output\n\nFor all $i$ from $1$ to $n$, print $t_i$.\n\n[samples]\n\n## Note\n\nThere are $3$ triples that have product $8$: $(1, 1, 8)$, $(1, 2, 4)$, and $(2, 2, 2)$. However, there is only $1$ triple that has product $7$: $(1, 1, 7)$.","is_translate":false,"language":"English"},{"statement_type":"Markdown","content":"**Definitions**  \nLet $ n \\in \\mathbb{Z}^+ $ with $ 1 \\leq n \\leq 10^4 $.  \nFor each $ i \\in \\{1, 2, \\dots, n\\} $, define $ t_i $ as the number of unordered triples $ (a, b, c) \\in \\mathbb{Z}^+^3 $ such that $ a \\leq b \\leq c $ and $ a \\cdot b \\cdot c = i $.\n\n**Objective**  \nFor each $ i = 1 $ to $ n $, compute and output $ t_i $.","is_translate":false,"language":"Formal"}],"meta":{"iden":"CF10264A2","tags":[],"sample_group":[],"created_at":"2026-03-03 11:00:39"}}