A363863 Numbers expressible as j^2 - k^2, 1 <= k <= j-2 ("squares with a square hole").

8, 12, 15, 16, 20, 21, 24, 27, 28, 32, 33, 35, 36, 39, 40, 44, 45, 48, 51, 52, 55, 56, 57, 60, 63, 64, 65, 68, 69, 72, 75, 76, 77, 80, 81, 84, 85, 87, 88, 91, 92, 93, 95, 96, 99, 100, 104, 105, 108, 111, 112, 115, 116, 117, 119, 120, 123, 124, 125, 128, 129, 132, 133, 135, 136

Offset: 1

Author

Thomas A. Fisher, Jun 25 2023

Keywords

• easy
• nonn

Comments

Inspired by my 4-year-old son, who loves Numberblocks, I decided to work out which numbers appear in the "squares with [square] holes club". These are numbers which, when configured as a square, have a square wholly removed. For example, 8 is 3 X 3 with a 1 X 1 hole in the middle. 24 is both a 5 X 5 with a 1 X 1 hole in the middle and a 7 X 7 with a 5 X 5 hole in the middle. The hole has to be "wholly contained", meaning I can't, for example, have 3^2 - 2^2 = 9 - 4 = 5, as removing a 2 X 2 square from a 3 X 3 square doesn't leave a "hole", as we are working with blocks, i.e., integers.

This sequence contains all natural numbers which factor as (j - k)*(j + k), where j - k >= 2 and k >= 1. That is, all natural numbers which have at least one factor pair of the form u*v such that u and v have the same parity, are distinct, and are both strictly greater than 1. This precisely rules out 1, primes, squares of primes, and the even numbers which are congruent to 2 modulo 4. In other words, this sequence is equal to A080257\A016825.

Examples

			8 = 3^2 - 1^2, 12 = 4^2 - 2^2, 15 = 4^2 - 1^2, ...
		

Crossrefs

Subsequence of A024352.

Equals A080257\A016825.

Programs

PARI

isok(k) = ((omega(k)>1) || (isprimepower(k)>2)) && ((k % 4) != 2); \\ Michel Marcus, Jun 30 2023