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A208323
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"Natural" rectangular pyramidal numbers.
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1
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1, 2, 3, 5, 5, 8, 7, 11, 14, 14, 11, 20, 13, 20, 26, 30, 17, 32, 19, 40, 38, 32, 23, 50, 55, 38, 50, 60, 29, 70, 31, 70, 62, 50, 85, 91, 37, 56, 74, 100, 41, 112, 43, 100, 115, 68, 47, 133, 140, 130, 98, 120, 53, 154, 145, 168, 110, 86, 59, 175, 61, 92, 196
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OFFSET
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1,2
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COMMENTS
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This sequence shows the maximum number of spheres in a pyramid with a rectangular base, where the base consists of n spheres. The area of the base n is the product of the lengths of its edges a and b, where 0 <= b <= a. In order to find the maximum number of spheres in the pyramid a(n), for a certain n we have to find factors a and b as close to each other, i.e. as close to sqrt(n), as possible. Therefore, b = A033676(n). The number b also represents the number of floors in the pyramid (i.e., its height in spheres).
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LINKS
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FORMULA
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a(n) = Sum_{i=0..b-1} (a-i)*(b-i), n=ab, 0 <= b <= a, b = A033676(n).
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EXAMPLE
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For n = a*b = 12, a and b must be as close to sqrt(12) as possible. Therefore, a=4, b=3 and a(n) = Sum_{i=0..2} (4-i)*(3-i)) = 20.
For any prime number n, a(n) = n.
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MATHEMATICA
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Table[If[IntegerQ[Sqrt[n]], w = h = Sqrt[n], d = Divisors[n]; len = Length[d]/2; {w, h} = d[[{len, len+1}]]]; Sum[(w - i) (h - i), {i, 0, w - 1}], {n, 63}] (* T. D. Noe, Feb 28 2012 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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