{
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  "body": "\n## Definition\n\nLet $n$ be a positive integer with [[prime factorization|0AEKDFCYFA]] $n = p_1^{e_1} \\cdots p_k^{e_k}$, with distinct primes $p_i$.\nThen $\\Omega(n) = e_1 + e_2 + \\dots + e_k$.\nSo $\\Omega(n)$ is the number of prime factors of $n$, counted with multiplicity.\n\n## Remarks\n\nFor example, $\\Omega(1) = 0$, $\\Omega(12) = 3$ and $\\Omega(360) = 6$.\nThe function $\\Omega$ is completely additive: $\\Omega(mn) = \\Omega(m) + \\Omega(n)$ for all positive $m$ and $n$.\nThe [[Liouville function|08RQAPXF1T]] is $\\lambda(n) = (-1)^{\\Omega(n)}$.\n\n## Explanations\n\n### L1 human\n\nThe function $\\Omega(n)$ counts all the prime factors of $n$, with repeats.\nFor $12 = 2 \\cdot 2 \\cdot 3$, there are three factors, so $\\Omega(12) = 3$, while $\\omega(12) = 2$.\n",
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