Sum of Factors of an Integer
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- The sum of factors of a positive integer is an arithmetic function, denoted by σ(n)\sigma(n)σ(n). For example, σ(1)=1\sigma(1)=1σ(1)=1 and σ(6)=6+3+2+1=12\sigma(6)=6+3+2+1=12σ(6…
By Kada Williams, Sean Li, Alex Federspiel
- Finding the sum of the factors of a positive integer is a very useful an interesting endeavor. Similar to other problems involving factors, it is often possible to find the answ…
By Trent Tormoehlen
- A perfect number is a positive integer nnn satisfying the equation σ(n)=2n\sigma(n)=2nσ(n)=2n. For example, 666 is a perfect number because 1+2+3=61+2+3=61+2+3=6, or 282828 is a…
By Kada Williams
- Using the formulas for σ\sigmaσ and φ\varphiφ (see here), prove that φ(n)σ(n)≤n2\varphi(n)\sigma(n)\le n^2φ(n)σ(n)≤n2 for every positive integer nnn.
By Kada Williams
- Prove the formula σk(n)=∏i=1rpik(αi+1)−1pik−1,\sigma_k(n)=\prod_{i=1}^r \frac{{p_i}^{k(\alpha_i+1)}-1}{{p_i}^k-1},σk(n)=i=1∏rpik−1pik(αi+1)−1, where kkk is a positive inte…
By Kada Williams