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A228729 Product of the positive squares less than or equal to n. 0

%I

%S 1,1,1,4,4,4,4,4,36,36,36,36,36,36,36,576,576,576,576,576,576,576,576,

%T 576,14400,14400,14400,14400,14400,14400,14400,14400,14400,14400,

%U 14400,518400,518400,518400,518400,518400,518400,518400,518400,518400,518400

%N Product of the positive squares less than or equal to n.

%C Squares of A214080, n > 0. Also, the n-th value of A001044 (The squared factorial numbers) repeated 2n+1 times, n > 0.

%C 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 a(n) = Product_{i=1..n} i^(1 - ceiling(frac(sqrt(i)))).

%F a(n) = A214080(n)^2, n > 0.

%e 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 seq(product( (i)^(1 - ceil(sqrt(i)) + floor(sqrt(i))), i = 1..k ), k=1..100);

%t Table[Times@@(Range[Floor[Sqrt[n]]]^2), {n, 50}] (* _Alonso del Arte_, Sep 01 2013 *)

%Y Cf. A000290, A001044, A010052, A214080.

%K nonn,easy,less

%O 1,4

%A _Wesley Ivan Hurt_, Aug 31 2013

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Last modified July 12 08:23 EDT 2020. Contains 335657 sequences. (Running on oeis4.)