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A351179
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Least positive integer m such that m^6*n = w^6 + x^3 + y^3 + z^3 for some nonnegative integers w,x,y,z.
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2
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1, 1, 1, 1, 1, 3, 5, 3, 1, 1, 1, 1, 5, 3, 3, 3, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 3, 1, 1, 1, 1, 6, 3, 3, 3, 1, 1, 1, 5, 3, 3, 3, 3, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 6, 3, 2, 2, 1, 1, 1, 3, 3, 3, 3, 3, 1, 1, 1, 2, 2, 2, 3, 3, 1, 2, 3, 1, 1, 1, 3, 3, 7, 3, 2, 1, 1
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OFFSET
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0,6
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COMMENTS
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a(n) always exists, because any positive rational number can be written as a sum of three cubes of positive rational numbers (see Richmond reference).
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th Edition, Oxford Univ. Press, 1960. (See Theorem 234 on page 197.)
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LINKS
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FORMULA
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EXAMPLE
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a(5) = 3 with 3^6*5 = 2^6 + 5^3 + 12^3 + 12^3.
a(12) = 5 with 5^6*12 = 3^6 + 19^3 + 34^3 + 52^3.
a(22) = 2 with 2^6*22 = 1^6 + 4^3 + 7^3 + 10^3.
a(31) = 6 with 6^6*31 = 0^6 + 4^3 + 15^3 + 113^3.
a(96) = 7 with 7^6*96 = 0^6 + 2^3 + 38^3 + 224^3.
a(101) = 4 with 4^6*101 = 3^6 + 22^3 + 39^3 + 70^3.
a(850) = 8 with 8^6*850 = 5^6 + 508^3 + 442^3 + 175^3.
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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-w^6-x^3-y^3], tab=Append[tab, m]; Goto[aa]], {w, 0, (k*n)^(1/6)}, {x, 0, ((k*n-w^6)/3)^(1/3)}, {y, x, ((k*n-w^6-x^3)/2)^(1/3)}];
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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