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 A253809 Array of pairs (x,y) of Markoff triples (x,y,z) with x <= y <= z, for z given in A002559. 0
 1, 1, 1, 1, 1, 2, 1, 5, 2, 5, 1, 13, 1, 34, 2, 29, 5, 13, 1, 89, 5, 29, 1, 233, 2, 169, 13, 34, 1, 610, 5, 194, 1, 1597, 2, 985, 5, 433, 13, 194, 34, 89, 1, 4181, 29, 169, 1, 10946, 2, 5741, 29, 433, 5, 2897, 13, 1325, 89, 233, 1, 28657 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Frobenius' conjecture on Markoff triples is that the maximal member z of the triple of positive integers (x,y,z), satisfying x^2 + y^2 + z^2 - 3*x*y*z = 0, with x <= y <= z, determines x and y uniquely. Also, each entry from A002559 (Markoff numbers) is conjectured to appear as a maximal member z. If an entry A002559(n) should not appear as z then one puts z(n) = 0 and row n will be 0, 0. If this Frobenius conjecture is true then the row length of this array is always 2, and only positive numbers appear. REFERENCES R. A. Mollin, Advanced Number Theory with Applications, Chapman & Hall/CRC, Boca Raton, 2010, 123-125. See also A002559. LINKS Table of n, a(n) for n=1..60. Feng-Juan Chen and Yong-Gao Chen, On the Frobenius conjecture for Markoff numbers, J. Number Theory 133 (2013) 2363-2373. Don Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation 39:160 (1982), pp. 709-723. See also A002559. EXAMPLE The array A(n,k) begins: If the Frobenius conjecture is true there will only be one pair x(1,n), y(1,n) for each z(n). n z(n) \ k=1: x(1,n) k=2: y(1,n) ... 1 1: 1 1 2 2: 1 1 3 5: 1 2 4 13: 1 5 5 29: 2 5 6 34: 1 13 7 89: 1 34 8 169: 2 29 9 194: 5 13 10 233: 1 89 11 433: 5 29 12 610: 1 233 13 985: 2 169 14 1325: 13 34 15 1597: 1 610 16 2897: 5 194 17 4181: 1 1597 18 5741: 2 985 19 6466: 5 433 20 7561: 13 194 21 9077: 34 89 22 10946: 1 4181 23 14701: 29 169 24 28657: 1 10946 25 33461: 2 5741 26 37666: 29 433 27 43261: 5 2897 28 51641: 13 1325 29 62210: 89 233 30 75025: 1 28657 ... CROSSREFS Cf. A002559. Sequence in context: A093664 A298212 A087620 * A262213 A178913 A142721 Adjacent sequences: A253806 A253807 A253808 * A253810 A253811 A253812 KEYWORD nonn,tabf AUTHOR Wolfdieter Lang, Jan 28 2015 STATUS approved

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Last modified June 3 04:09 EDT 2023. Contains 363103 sequences. (Running on oeis4.)