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  "body": "\n## Definition\n\nFor a positive integer $n$, $\\omega(n)$ is the number of distinct [[prime numbers|0SN0MAEKYX]] that divide $n$.\nSo if $n = p_1^{e_1} \\cdots p_k^{e_k}$ with distinct primes $p_i$ and $e_i \\ge 1$, then $\\omega(n) = k$.\n\n## Remarks\n\nFor example, $\\omega(1) = 0$, $\\omega(12) = 2$ and $\\omega(30) = 3$.\nThe function $\\omega$ is additive: $\\omega(mn) = \\omega(m) + \\omega(n)$ for coprime $m$ and $n$.\nCompare with the [[number of prime factors with multiplicity|08420828EY]] $\\Omega(n)$. We have $\\omega(n) \\le \\Omega(n)$, with equality exactly for squarefree $n$.\n\n## Explanations\n\n### L1 human\n\nThe function $\\omega(n)$ counts the different primes in $n$ and ignores repeats.\nFor $360 = 2^3 \\cdot 3^2 \\cdot 5$, the different primes are $2$, $3$ and $5$, so $\\omega(360) = 3$.\n",
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