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 A228729 Product of the positive squares less than or equal to n. 0
 1, 1, 1, 4, 4, 4, 4, 4, 36, 36, 36, 36, 36, 36, 36, 576, 576, 576, 576, 576, 576, 576, 576, 576, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 518400, 518400, 518400, 518400, 518400, 518400, 518400, 518400, 518400, 518400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Squares of A214080, n > 0. Also, the n-th value of A001044 (The squared factorial numbers) repeated 2n+1 times, n > 0. 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)). LINKS FORMULA a(n) = Product_{i=1..n} i^(1 - ceiling(frac(sqrt(i)))). a(n) = A214080(n)^2, n > 0. EXAMPLE a(6) = 4 since there are two squares less than or equal to 6 (1 and 4) and their product is 1*4 = 4. MAPLE seq(product( (i)^(1 - ceil(sqrt(i)) + floor(sqrt(i))), i = 1..k ), k=1..100); MATHEMATICA Table[Times@@(Range[Floor[Sqrt[n]]]^2), {n, 50}] (* Alonso del Arte, Sep 01 2013 *) CROSSREFS Cf. A000290, A001044, A010052, A214080. Sequence in context: A035627 A228423 A165923 * A174444 A268237 A276866 Adjacent sequences:  A228726 A228727 A228728 * A228730 A228731 A228732 KEYWORD nonn,easy,less AUTHOR Wesley Ivan Hurt, Aug 31 2013 STATUS approved

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Last modified May 29 06:31 EDT 2020. Contains 334697 sequences. (Running on oeis4.)