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Search: id:A083356
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| A083356 |
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Total area of all incongruent integer-sided rectangles of area <= n. |
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+0
4
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| 0, 1, 3, 6, 14, 19, 31, 38, 54, 72, 92, 103, 139, 152, 180, 210, 258, 275, 329, 348, 408, 450, 494, 517, 613, 663, 715, 769, 853, 882, 1002, 1033, 1129, 1195, 1263, 1333, 1513, 1550, 1626, 1704, 1864, 1905, 2073, 2116, 2248, 2383, 2475, 2522, 2762, 2860
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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Nick MacKinnon, Problem 10883, Amer. Math. Monthly, 108 (2001) 565; solution by John C. Cock, 110 (2003) 343-344.
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FORMULA
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a(n) = sum_{k=1..n} k*ceiling(d(k)/2), where d(k)=A000005(k) is the number of divisors of k.
a(n) = sum_{r=1..floor(sqrt(n))} r*(r+floor(n/r))*(floor(n/r)+1-r)/2.
a(n) ~ n^2 * log(n) / 4
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EXAMPLE
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a(5)=19, the rectangles being 1 X 1, 1 X 2, 1 X 3, 1 X 4, 1 X 5 and 2 X 2.
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MATHEMATICA
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a[n_] := Sum[r(r+Floor[n/r])(Floor[n/r]+1-r), {r, 1, Floor[Sqrt[n]]}]/2
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CROSSREFS
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Cf. A083357. Partial sums of A060872.
Sequence in context: A075390 A118523 A097633 this_sequence A096337 A109757 A075189
Adjacent sequences: A083353 A083354 A083355 this_sequence A083357 A083358 A083359
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KEYWORD
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nonn,easy
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AUTHOR
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Dean Hickerson (dean.hickerson(AT)yahoo.com), Apr 26 2003
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