Neat integer sequences

oeis.org/A099155

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The 39 non-rectangular nonominoes that can tile a rectangle with copies of itself.

Here's the other 39, and they all have an order of 2 or 4 (from cflmath.com/Polyomino/re... ):

A126140(9) = 4 is surprising! There are 1,285 free nonominoes (ignoring rotations and reflections), and 1,050 can tile the plane, but only 41 can tile a rectangle, and two of those are trivial: oeis.org/A126138

Interesting thread in which oeis.org/A126140 makes a surprise appearance

And the one after that is 16,208 digits long. (It ends with a 7.)

oeis.org/A060421

I'm not going to keep going because the next prime that's a prefix of pi is 31415926535897932384626433832795028841

3 is prime
31 is prime
314 is not, obviously (2 x 157)
3141 = 3 x 3 x 349
31415 = 5 x 61 x 103
314159 is prime!

A005042: Primes formed by the initial digits of the decimal expansion of Pi

oeis.org/A005042

Probably my favorite variant is bimagic squares, where if you square every value in a magic square it's still magic. It's impossible to make a bimagic square smaller than 8x8. www.multimagie.com/English/Smal...

There are many sub-categories of magic squares, like associative (every pair of numbers symmetrically opposite the center sum to the same value) and panmagic (where diagonals that wrap around also sum to the same number as the two standard diagonals)

The number of squares for n=6 was only calculated in 2023: magicsquare6.net

The Luoshu magic square, a 3x3 grid of numbers where the rows, columns, and diagonals all add up to 15

The Luoshu Square, the only solution for n=3, has been certainly known since the 6th century, and possibly hundreds of years longer than that.

en.wikipedia.org/wiki/Luoshu_...

Magic squares are square grids where each cell is filled with a number from 1 to the number of cells, and where every row, column, and diagonal sums to the same value.

A006052: Number of magic squares of order n composed of the numbers from 1 to n^2

1, 0, 1, 880, 275305224, 17753889197660635632

oeis.org/A081263