This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A228729 #20 Aug 15 2025 12:16:05
%S A228729 1,1,1,4,4,4,4,4,36,36,36,36,36,36,36,576,576,576,576,576,576,576,576,
%T A228729 576,14400,14400,14400,14400,14400,14400,14400,14400,14400,14400,
%U A228729 14400,518400,518400,518400,518400,518400,518400,518400,518400,518400,518400
%N A228729 Product of the positive squares less than or equal to n.
%C A228729 Squares of A214080, n > 0. Also, the n-th value of A001044 (The squared factorial numbers) repeated 2n+1 times, n > 0.
%C A228729 The first differences of a(n) are positive when n is a square (i.e., a(n+1) - a(n) > 0) and zero otherwise. This implies that the square characteristic (A010052) can be written in terms of a(n) as A010052(n) = signum(a(n+1) - a(n)), n > 1. Furthermore, the number of squares less than or equal to n is given by Sum_{i=1..n} sign(a(i+1) - a(i)), and the sum of the squares less than or equal to n is given by Sum_{i=2..n} i * sign(a(i+1) - a(i)).
%F A228729 a(n) = Product_{i=1..n} i^(1 - ceiling(frac(sqrt(i)))).
%F A228729 a(n) = A214080(n)^2, n > 0.
%F A228729 Sum_{n>=1} 1/a(n) = BesselI(0, 2) + 2*BesselI(1, 2) - 1. - _Amiram Eldar_, Aug 15 2025
%e A228729 a(6) = 4 since there are two squares less than or equal to 6 (1 and 4) and their product is 1*4 = 4.
%p A228729 seq(product( (i)^(1 - ceil(sqrt(i)) + floor(sqrt(i))), i = 1..k ), k=1..100);
%t A228729 Table[Times@@(Range[Floor[Sqrt[n]]]^2), {n, 50}] (* _Alonso del Arte_, Sep 01 2013 *)
%Y A228729 Cf. A000290, A001044, A010052, A214080.
%K A228729 nonn,easy,less
%O A228729 1,4
%A A228729 _Wesley Ivan Hurt_, Aug 31 2013