A329402 Number of rectangles (w X h, w <= h) with integer side lengths w and h having area = n * perimeter.
2, 3, 5, 4, 5, 8, 5, 5, 8, 8, 5, 11, 5, 8, 14, 6, 5, 13, 5, 11, 14, 8, 5, 14, 8, 8, 11, 11, 5, 23, 5, 7, 14, 8, 14, 18, 5, 8, 14, 14, 5, 23, 5, 11, 23, 8, 5, 17, 8, 13, 14, 11, 5, 18, 14, 14, 14, 8, 5, 32, 5, 8, 23, 8, 14, 23, 5, 11, 14, 23, 5, 23, 5, 8, 23
Offset: 1
Keywords
Examples
a(1) = 2 because there are two rectangles which have area = perimeter: 4 X 4 and 3 X 6. a(2) = 3 because there are 3 rectangles for which area = 2 * perimeter: 8 X 8, 6 X 12, and 5 X 20. a(3) = 5 because there are 5 rectangles for which area = 3 * perimeter: 12 X 12, 10 X 15, 9 X 18, 8 X 24, and 7 X 42.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- B. Greer, D. De Bock and W. Van Dooren, The Isis Problem as an Experimental probe and teaching resource, Journal of Mathematical Behavior, 28, (2009), 237-246.
Crossrefs
Cf. A038548.
Programs
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Maple
f:= n -> (numtheory:-tau(4*n^2)+1)/2; map(f, [$1..100]); # Robert Israel, Mar 31 2020
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Mathematica
a[n_] := Ceiling[DivisorSigma[0, 4 n^2]/2]; Array[a, 75] (* Giovanni Resta, Mar 29 2020 *)
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Python
numbers=[] for n in range(500): c=int(0) n=int(n+1) for x in range(2*n+1,4*n+1): y=(2*n*x)/(x-2*n) if y==y//1: y=int(y) c=c+1 numbers.append(c) print(numbers)
Formula
a(n) = A038548(4*n^2). - Peter Bala, Mar 03 2020
Comments