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A336332
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Largest side, in increasing order, of primitive integer-sided triangles with A < B < C < 2*Pi/3 and such that FA + FB + FC is an integer where F is the Fermat point of the triangle.
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10
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73, 95, 152, 205, 208, 280, 296, 343, 361, 387, 407, 437, 469, 473, 485, 624, 728, 931, 1016, 1273, 1311, 1313, 1368, 1387, 1443, 1457, 1463, 1469, 1477, 1519, 1560, 1591, 1687, 1895, 2015, 2045, 2045, 2085, 2197, 2231, 2289, 2347, 2363, 2416, 2465, 2553, 2728, 2821, 2923
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OFFSET
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1,1
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COMMENTS
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Inspired by Project Euler, Problem 143 (see link).
This sequence is increasing because triples are in increasing order of largest side.
For the corresponding primitive triples and miscellaneous properties and references, see A336328.
If FA + FB + FC = d, then we have this "beautifully symmetric equation" between a, b, c and d (see Martin Gardner):
3*(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.
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REFERENCES
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Martin Gardner, Mathematical Circus, Elegant triangles, First Vintage Books Edition, 1979, p. 65
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LINKS
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FORMULA
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EXAMPLE
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a(36) = a(37) = 2045 is the smallest largest side that appears twice because:
(1023, 1387, 2045) is a triple with FA+FB+FC = 2408, and
(1051, 1744, 2045) is a triple with FA+FB+FC = 2709.
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CROSSREFS
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Cf. A072052 (largest sides: primitives and multiples).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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