ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Prime-power formulas for $τ$ and $σ_{k}$ in concrete cases

Example

Using the prime-power formulas from The divisor functions arise by Dirichlet convolution:

τ (12) = (2 + 1) (1 + 1) = 6, σ (12) = (1 + 2 + 4) (1 + 3) = 28,

since $12 = 2^{2} \cdot 3$ , and

τ (18) = (1 + 1) (2 + 1) = 6, σ_{2} (18) = (1 + 2^{2}) (1 + 3^{2} + 3^{4}) = 455,

since $18 = 2 \cdot 3^{2}$ .

Facts & Assumptions

Given: The integers $12$ and $18$ .

Verification

technique · direct

1.1

By For $n \geq 1$ and any injective list $p : r \to Z$ of primes containing every prime divisor of $n$ , one has $n = \prod_{i < r} p_{i}^{v_{p_{i}} (n)}$ ; the exponents are determined by $n$ , and $v_{q} (n) = 0$ for every prime $q$ outside the list, $12 = 2^{2} \cdot 3$ and $18 = 2 \cdot 3^{2}$ . Applying the prime-power formulas of The divisor functions arise by Dirichlet convolution gives the displayed products for $τ (12)$ , $σ (12)$ , $τ (18)$ , and $σ_{2} (18)$ .

given

2.1

Evaluating those products yields $τ (12) = 6$ , $σ (12) = 7 \cdot 4 = 28$ , $τ (18) = 6$ , and $σ_{2} (18) = 5 \cdot 91 = 455$ .

step 1.1algebra

3.1

The direct divisor lists ${1, 2, 3, 4, 6, 12}$ and ${1, 2, 3, 6, 9, 18}$ have six elements each, and $1 + 2 + 3 + 4 + 6 + 12 = 28$ while $1 + 2^{2} + 3^{2} + 6^{2} + 9^{2} + 18^{2} = 455$ , so the prime-power formulas agree with direct computation in these cases.

step 2.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Victor Shoup, A Computational Introduction to Number Theory and Algebra, Sections 2.9 and 2.10 (standard reference, not scraped)

Prime-power formulas for ττ and σkσk​ in concrete cases