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 A156688 The total number of distinct Pythagorean triples with an area numerically equal to n times their perimeters 2
 2, 3, 6, 4, 6, 9, 6, 5, 10, 9, 6, 12, 6, 9, 18, 6, 6, 15, 6, 12, 18, 9, 6, 15, 10, 9, 14, 12, 6, 27, 6, 7, 18, 9, 18, 20, 6, 9, 18, 15, 6, 27, 6, 12, 30, 9, 6, 18, 10, 15, 18, 12, 6, 21, 18, 15, 18, 9, 6, 36, 6, 9, 30, 8, 18, 27, 6, 12, 18, 27, 6, 25, 6, 9, 30, 12, 18, 27, 6, 18, 18, 9, 6, 36, 18, 9, 18, 15, 6, 45, 18, 12, 18, 9, 18 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The members of this sequence are also 1/2 the number of divisors of 8n^2. The corresponding results for primitive triangles only are in A068068. Also, the total number of distinct "areas with equal border", that is: Let x, y be positive integers so that the area xy equals the border around it with thickness n. As a formula it is: 2xy = (x+2n)(y+2n). To compare with the original, the areas at thickness 5 are 11x210, 12x110, 14x60, 15x50, 18x35, 20x30. - Juhani Heino, Jul 22 2012 REFERENCES Chi, Henjin and Killgrove, Raymond; Problem 1447, Crux Math 15(5), May 1989. Chi, Henjin and Killgrove, Raymond; Solution to Problem 1447, Crux Math 16(7), September 1990. LINKS Antti Karttunen, Table of n, a(n) for n = 1..65537 Ron Knott, Right-angled Triangles and Pythagoras' Theorem FORMULA a(n) = A000005(8n^2)/2 = A078644(2n). EXAMPLE There are 6 Pythagorean triples whose area is 5 times their perimeters - (21,220,221), (22,120,122), (24,70,74), (25,60,65),(28,45,53) and (30,40,50) - hence a(5)=6. MATHEMATICA 1/2 DivisorSigma[0, 8#^2] &/@Range[75] PROG (PARI) A156688(n) = (numdiv(8*n*n)/2); \\ Antti Karttunen, Sep 27 2018 CROSSREFS Cf. A000005, A068068. Sequence in context: A306231 A125703 A349381 * A348243 A019567 A098286 Adjacent sequences: A156685 A156686 A156687 * A156689 A156690 A156691 KEYWORD easy,nice,nonn AUTHOR Ant King, Feb 18 2009 EXTENSIONS More terms from Antti Karttunen, Sep 27 2018 STATUS approved

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Last modified June 6 12:57 EDT 2023. Contains 363142 sequences. (Running on oeis4.)