OEIS A000203: Sum of divisors

Sigma(n) is the sum of all positive divisors of n (including 1 and n itself). It is multiplicative: if gcd(a,b)=1 then sigma(ab)=sigma(a)sigma(b), and for a prime power p^a we have sigma(p^a)=1+p+p^2+...+p^a=(p^{a+1}-1)/(p-1). Thus sigma(n) for any n with prime factorization n=∏p_i^{a_i} is the product sigma(n)=∏(p_i^{a_i+1}-1)/(p_i-1). We'll walk through examples like n=6 (sigma=12) and n=12 (sigma=28), compare with the divisor-counting function A000005 and the aliquot sum A001065, and discuss how the sum-of-divisors classifies numbers as perfect, abundant, or deficient. A key fact: sigma(n) is odd iff n is a square or twice a square. We’ll also touch connections to lattices and groups (counting sublattices of index n, etc.), the Dirichlet generating function sigma(n) ~ zeta(s)zeta(s-1), and common clarifications (for example, not all “Euler-type” recurrences are for sigma).

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