Divisor Sum Calculator – σ(n) Function

The divisor function σ(n) adds every positive divisor of n. For 12, the divisors 1, 2, 3, 4, 6, 12 sum to σ(12) = 28. The prime-factorization shortcut computes it without listing: 12 = 2²×3 gives σ = (1+2+4)(1+3) = 7×4 = 28, using the geometric-series rule σ(pᵃ) = 1 + p + … + pᵃ = (pᵃ⁺¹−1)/(p−1). Multiplicativity across coprime parts lets σ(10) = σ(2)·σ(5) = 3×6 = 18. The calculator shows the divisor list with running sum, the factorized computation, and the derived quantities: the aliquot sum σ(n) − n (16 for 12, flagging it abundant), and the perfect-number test σ = 2n (σ(28) = 56 ✓, σ(6) = 12 ✓). It also reports τ(n), the divisor count — τ(12) = 6 — and notes squares have odd τ.

1. 1

   Enter n

   Any positive integer

2. 2

   List the divisors

   All d | n with a running total

3. 3

   Factor shortcut

   (1+2+4)(1+3) = 28 recomputes σ(12)

4. 4

   Classify

   Perfect (σ = 2n), abundant (σ > 2n), or deficient (σ < 2n)

• 1

  σ(pᵃ) = (pᵃ⁺¹−1)/(p−1) — the geometric series rule

• 2

  Perfect ⇔ σ(n) = 2n; aliquot sum = σ(n) − n

• 3

  τ(n) counts divisors; squares have odd τ

• Omitting n itself from the divisor list

• Multiplying σ values for non-coprime parts (must factor first)

• Confusing σ with the aliquot sum σ − n

• Missing the factor (p⁰ = 1) in the geometric series