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A351321
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Least positive integer m such that m^6*n = u^6 + v^3 - (x^6 + y^3) for some nonnegative integers u,v,x,y with x^6 + y^3 <= m^6*n^2.
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4
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1, 1, 1, 21, 6, 3, 1, 1, 1, 1, 7, 7, 3, 3, 3, 3, 2, 6, 1, 1, 1, 2, 6, 3, 5, 1, 1, 1, 1, 5, 2, 6, 12, 3, 1, 1, 1, 1, 1, 6, 6, 3, 3, 2, 1, 1, 4, 3, 2, 3, 3, 2, 7, 1, 1, 1, 1, 1, 3, 6, 1, 1, 1, 1, 1, 1, 4, 15, 3, 3, 1, 1, 1, 3, 4, 2, 3, 6, 3, 3, 2, 3, 1, 1, 3, 3, 3, 6, 1, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1
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OFFSET
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0,4
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COMMENTS
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6-6-3-3 Conjecture: Each rational number can be written as u^6 - v^6 + x^3 - y^3 with u,v,x,y nonnegative rational numbers. Moreover, a(n) exists for any nonnegative integer n.
As a/b = (a*b^5)/b^6 for any integer a and nonzero integer b, the second assertion in the conjecture implies the first one.
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LINKS
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EXAMPLE
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a(3) = 21 with 21^6*3 = 22^6 + 956^3 - (30^6 + 93^3) and 30^6 + 93^3 <= 21^6*3^2.
a(67) = 15 with 15^6*67 = 21^6 + 1091^3 - (15^6 + 848^3) and 15^6 + 848^3 <= 15^6*67^2.
a(564) = 14 with 14^6*564 = 69^6 + 4415^3 - (16^6 + 5746^3) and 16^6 + 5746^3 <= 14^6*564^2.
a(949) = 18 with 18^6*949 = 7^6 + 11784^3 - (11^6 + 11706^3) and 11^6 + 11706^3 <= 18^6*949^2.
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MATHEMATICA
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CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={}; Do[m=1; Label[bb]; k=m^6; Do[If[CQ[k*n+x^6+y^3-z^6], tab=Append[tab, m]; Goto[aa]],
{x, 0, m*n^(1/3)}, {y, 0, (k*n^2-x^6)^(1/3)}, {z, 0, (k*n+x^6+y^3)^(1/6)}]; m=m+1; Goto[bb]; Label[aa], {n, 0, 100}]; Print[tab]
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CROSSREFS
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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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