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A060466
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Value of y of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest |z| and smallest |y|, 0 <= |x| <= |y| <= |z|.
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5
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0, 0, 1, 1, -1, -1, 0, 1, 1, -2, 10, 2, -1609, 2, -2, -2, -2, -14, -15550555555, -1, -1, 0, 1, 1, -2218888517, -8778405442862239, 2, 2, 2, -3, -3, 134476, 80435758145817515, 2, -7, -3, 3, 7, -26, 659, 60702901317, 3, -11, 3, -21, -2, -4, -4, 3, -1, 0, 1, 1, -4, 20, 2, 9, 2
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OFFSET
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0,10
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COMMENTS
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Only primitive solutions where gcd(x,y,z) does not divide n are considered.
From the solution A060464(24) = 30 = -283059965^3 - 2218888517^3 + 2220422932^3 (smallest possible magnitudes according to A. Bogomolny), one has a(24) = -2218888517. A solution to A060464(25) = 33 remains to be found. Other values for larger n can be found in the second column of the table on Hisanori Mishima's web page. - M. F. Hasler, Nov 10 2015
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, Section D5.
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LINKS
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EXAMPLE
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For n = 16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term -1609.
42 = 12602123297335631^3 + 80435758145817515^3 + (-80538738812075974)^3 was found by Andrew Booker and Andrew Sutherland.
74 = 66229832190556^3 + 283450105697727^3 + (-284650292555885)^3 was found by Sander Huisman.
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MATHEMATICA
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nmax = 29; A060464 = Select[Range[0, nmax], Mod[#, 9] != 4 && Mod[#, 9] != 5 &]; A060465 = {0, 0, 0, 1, -1, 0, 0, 0, 1, -2, 7, -1, -511, 1, -1, 0, 1, -11, -2901096694, -1, 0, 0, 0, 1}; r[n_, x_] := Reduce[0 <= Abs[x] <= Abs[y] <= Abs[z] && n == x^3 + y^3 + z^3, {y, z}, Integers]; A060466 = Table[y /. ToRules[ Simplify[ r[A060464[[k]], A060465[[k]]] /. C[1] -> 0]], {k, 1, Length[A060464]}] (* Jean-François Alcover, Jul 11 2012 *)
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CROSSREFS
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KEYWORD
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sign,nice,hard
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AUTHOR
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EXTENSIONS
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In order to be consistent with A060465, where only primitive solutions are selected, a(18)=2 was replaced with -15550555555, by Jean-François Alcover, Jul 11 2012
a(25) from Tim Browning and further terms added by Charlie Neder, Mar 09 2019
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STATUS
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approved
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