cp's OEIS Frontend

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

%I A363863 #45 May 31 2024 22:06:55
%S A363863 8,12,15,16,20,21,24,27,28,32,33,35,36,39,40,44,45,48,51,52,55,56,57,
%T A363863 60,63,64,65,68,69,72,75,76,77,80,81,84,85,87,88,91,92,93,95,96,99,
%U A363863 100,104,105,108,111,112,115,116,117,119,120,123,124,125,128,129,132,133,135,136
%N A363863 Numbers expressible as j^2 - k^2, 1 <= k <= j-2 ("squares with a square hole").
%C A363863 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.
%C A363863 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.
%e A363863 8 = 3^2 - 1^2, 12 = 4^2 - 2^2, 15 = 4^2 - 1^2, ...
%o A363863 (PARI) isok(k) = ((omega(k)>1) || (isprimepower(k)>2)) && ((k % 4) != 2); \\ _Michel Marcus_, Jun 30 2023
%Y A363863 Subsequence of A024352.
%Y A363863 Equals A080257\A016825.
%K A363863 easy,nonn
%O A363863 1,1
%A A363863 _Thomas A. Fisher_, Jun 25 2023