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A336330
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Smallest side 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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57, 73, 43, 127, 97, 111, 49, 95, 296, 152, 323, 147, 285, 255, 247, 469, 403, 871, 561, 657, 559, 1083, 833, 1057, 485, 507, 1072, 760, 767, 379, 211, 195, 1208, 952, 1443, 1023, 1051, 889, 1240, 1209, 1249, 1423, 1005, 1679, 1568, 1843, 193, 485, 1512
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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).
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.
This sequence is not increasing. For example, a(2) = 73 for triangle with largest side = 95 while a(3) = 43 for triangle with largest side = 152.
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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(1) = 57 because the first triple is (57, 65, 73) with corresponding d = FA + FB + FC = 264/7 + 195/7 + 325/7 = 112 and the symmetric relation satisfies: 3*(57^4 + 65^4 + 73^4 + 112^4) = (57^2 + 65^2 + 73^2 + 112^2)^2 = 642470409.
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CROSSREFS
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Cf. A072054 (smallest 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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