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  "body": "\n## Definition\n\nFor a positive integer $n$, the sum of divisors of $n$ is\n$$\\sigma(n) = \\sigma_1(n) = \\sum_{d \\mid n} d,$$\nwhere $\\sigma_1$ is the [[divisor function|0YEW47VJ0J]] with $k = 1$.\n\n## Remarks\n\nIf $n = p_1^{e_1} \\cdots p_r^{e_r}$ with distinct primes $p_i$, then $\\sigma(n) = \\prod_{i=1}^{r} \\frac{p_i^{e_i + 1} - 1}{p_i - 1}$.\nFor example, $\\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$.\nThe sum of the proper divisors is $\\sigma(n) - n$. It defines [[perfect|0SB7Q8SYSF]], [[abundant|0VS0MP1P5H]] and [[deficient|0G01TS1RRP]] numbers.\n\n## Explanations\n\n### L1 human\n\nThe function $\\sigma(n)$ adds all the divisors of $n$, including $1$ and $n$.\nFor a prime $p$, the only divisors are $1$ and $p$, so $\\sigma(p) = p + 1$.\n",
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