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A336330
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.
10
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
OFFSET
1,1
COMMENTS
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.
REFERENCES
Martin Gardner, Mathematical Circus, Elegant triangles, First Vintage Books Edition, 1979, p. 65.
FORMULA
a(n) = A336328(n, 1).
EXAMPLE
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.
CROSSREFS
Cf. A336328 (triples), A336329 (FA + FB + FC), this sequence (smallest side), A336331 (middle side), A336332 (largest side), A336333 (perimeter).
Cf. A072054 (smallest sides: primitives and multiples).
Sequence in context: A067809 A176636 A072054 * A230956 A039429 A043252
KEYWORD
nonn
AUTHOR
Bernard Schott, Jul 21 2020
STATUS
approved