Math 2.0

Number theory Definition

Sum of divisors

Builds on Divisor function

Definition

For a positive integer nn, the sum of divisors of nn is σ(n)=σ1(n)=∑d∣nd,\sigma(n) = \sigma_1(n) = \sum_{d \mid n} d, where σ1\sigma_1 is the divisor function with k=1k = 1.

Remarks

If n=p1e1⋯prern = p_1^{e_1} \cdots p_r^{e_r} with distinct primes pip_i, then σ(n)=∏i=1rpiei+1−1pi−1\sigma(n) = \prod_{i=1}^{r} \frac{p_i^{e_i + 1} - 1}{p_i - 1}. For example, σ(12)=1+2+3+4+6+12=28\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. The sum of the proper divisors is σ(n)−n\sigma(n) - n. It defines perfect, abundant and deficient numbers.

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Arithmetic function, definition, no evidence levelArithmetic functiondefinitionDivisibility, definition, no evidence levelDivisibilitydefinitionDivisors of a natural number, definition, no evidence levelDivisors of a natural…definitionDivisor function, definition, no evidence levelDivisor functiondefinitionSum of divisors, definition, no evidence levelSum of divisorsdefinition
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  1. Arithmetic function definition
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  3. Divisors of a natural number definition
  4. Divisor function definition
  5. Sum of divisors definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Divisor_function, Definition
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DefinesSum of divisors
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