Family of arithmetic functions σ_k(n) = ∑_{d|n} d^k that sum the k-th powers of positive divisors of n; the common special cases are σ_0(n)=d(n) (number of divisors) and σ_1(n)=σ(n) (sum of divisors).

Principle

Multiplicative for fixed k and computable from prime-power factorization by σ_k(p^a) = (p^{(a+1)k} − 1) / (p^k − 1). The σ_k encode how divisor structure weighted by powers aggregates across prime factors.

Demonstration

Example: n = 6 with divisors 1,2,3,6 gives σ_0(6)=4, σ_1(6)=1+2+3+6=12, σ_2(6)=1+4+9+36=50. For k = 1, σ(n) appears in the definition of perfect numbers (σ(n) = 2n).

Misapplication

Applying the prime-power formula without checking that k is fixed, using σ_k for non-integer k without clarifying meaning, or confusing σ_k with multiplicative functions of a different convolutional origin (e.g., Dirichlet convolutions that involve μ) leads to misuse.

Consequence

σ_k functions connect factorization to additive invariants, appear in divisor-sum identities, Euler products and Dirichlet series, and characterize special classes of integers (perfect, abundant, deficient) when k = 1.

Reversal

A complementary viewpoint is the sum of proper divisors s(n) = σ(n) − n; studying s(n) instead highlights deficiency or abundance. Another contrast is using indicator/count functions (d) rather than weighted sums (σ_k).

Boundary

Defined for positive integers n and usually integer k (often k ≥ 0); extensions to negative or complex k require analytic interpretation and care. Multiplicativity and the closed prime-power formula depend on arithmetic factorization.

Semantic Tension

Tension arises with the divisor-counting function d(n) (σ_0) and with multiplicative convolutions involving μ or other arithmetic functions; there is also tension between algebraic characterizations (perfect numbers) and analytic behaviors (average order and Dirichlet series).

Synthesis

The sigma family σ_k(n) uniformly packages divisor information by summing k-th powers of divisors; multiplicativity yields explicit prime-power formulae that link multiplicative factor structure to additive divisor sums, with k tuning the weighting of large versus small divisors.