Divisor functions \(d(n)\) and \(\sigma_k(n)\) refresher¶

Divisor functions measure how many divisors a number has, and how large they are. They are classical examples of multiplicative functions. See [Apostol, 1976].

Counting divisors: \(d(n)\)¶

Let \(d(n)\) (also written \(\tau(n)\)) be the number of positive divisors of \(n\).

If \(n=\prod p_i^{\alpha_i}\), then

\[ d(n)=\prod (\alpha_i+1). \]

Sum of divisors: \(\sigma_k(n)\)¶

For \(k\ge 0\),

\[ \sigma_k(n)=\sum_{d\mid n} d^k. \]

The case \(k=1\) is the usual sum-of-divisors function \(\sigma(n)\).

For a prime power,

\[ \sigma_k(p^\alpha)=1+p^k+p^{2k}+\cdots+p^{\alpha k} =\frac{p^{(\alpha+1)k}-1}{p^k-1}. \]

Again, multiplicativity gives \(\sigma_k(n)\) from prime powers.

Highly composite numbers (extremal behavior)¶

Numbers that maximize \(d(n)\) up to a range are called highly composite numbers. Ramanujan’s classic paper studies their structure. [Ramanujan, 1915]

Experiment ideas¶

visualize \(d(n)\) up to \(N\) and highlight record-breakers
compare \(\sigma(n)\) to \(n\) (abundant / perfect / deficient classification)
log-scale plots of \(d(n)\) to make extremes visible

Experiments in this repository¶

E096 — Record-holders for τ(n) up to N (highly composite flavor).
E097 — σ(n)/n landscape: deficient / perfect / abundant classification.
E098 — Extremals of σ(n)/n^α across α (phase changes / superabundant intuition).

References¶

See References.

[Alaoglu and Erdős, 1944, Apostol, 1976, Lagarias, 2002, Ramanujan, 1915, Robin, 1984]