Mathematical Explorations

Interactive visualizations and investigations in number theory and discrete structures.

OEIS Sequences

The On-Line Encyclopedia of Integer Sequences (OEIS) is a database of integer sequences. Below are my contributions:

A custom integer sequence counting primes in half-open ternary intervals.

Number Theory

The \(n\)-th term is defined as the number of primes in the half-open interval \([3^{n-1}, 3^n)\) for \(n > 0\).

Select Term (n)n = 3

Interval: [9, 27)

Primes Count: a(3) = 5

Growth log-scale (bars represent log10 of counts):

Primes in [9, 27) :

A custom integer sequence counting pairs with perfect square greatest common divisors.

Number Theory

The \(n\)-th term of this sequence counts the number of pairs \((x, y)\) with \(1 \le x < y \le n\) such that \(\gcd(x, y)\) is a perfect square.

Grid Limit (n)n = 60

Total Pairs (\(1 \le x < y \le n\)): 1,770

Square GCD Pairs: \(a(n)\) = 1,187

Proportion: 67.06%

Upper-triangular pair grid (\(1 \le x < y \le n\)):

Perfect Square GCD

Non-Square GCD

Hover over cells in the grid to inspect the pairs and their Greatest Common Divisors