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A343368
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Number of ways to write n as floor((a^3+b^3)/3) + floor((c^3+d^3)/5), where a,b,c,d are nonnegative integers with a > b and c >= d.
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7
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3, 2, 3, 6, 2, 3, 1, 3, 2, 7, 6, 3, 6, 2, 7, 2, 6, 1, 2, 2, 1, 10, 6, 3, 6, 6, 5, 6, 6, 4, 4, 5, 1, 4, 9, 6, 4, 4, 1, 5, 2, 4, 7, 5, 6, 5, 13, 6, 4, 6, 6, 7, 6, 5, 6, 8, 4, 4, 4, 5, 3, 2, 2, 4, 7, 4, 4, 8, 8, 5, 6, 6, 9, 8, 7, 8, 3, 15, 2, 10, 3, 8, 4, 3, 7, 6, 8, 4, 7, 9, 5, 4, 7, 8, 6, 6, 2, 8, 10, 4, 6
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OFFSET
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0,1
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COMMENTS
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Conjecture 1: a(n) > 0 for all n >= 0.
Conjecture 2: Each n = 0,1,... can be written as floor((a^3+b^3)/4) + floor((c^3+d^3)/5) with a,b,c,d nonnegative integers.
Both conjectures have been verified for all n = 0..10^5.
We also conjecture that the pair (4,5) of denominators in Conjecture 2 can be replaced by some other pairs such as (4,6), (5,6), (3,7), (4,7), (5,7), (6,7).
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LINKS
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EXAMPLE
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a(1) = 2 with 1 = floor((1^3+0^3)/3) + floor((2^3+0^3)/5) = floor((1^3+0^3)/3) + floor((2^3+1^3)/5).
a(17) = 1 with 17 = floor((2^3+1^3)/3) + floor((4^3+2^3)/5).
a(20) = 1 with 20 = floor((2^3+0^3)/3) + floor((4^3+3^3)/5).
a(38) = 1 with 38 = floor((4^3+2^3)/3) + floor((4^3+2^3)/5).
a(103) = 1 with 103 = floor((6^3+4^3)/3) + floor((3^3+3^3)/5).
a(304) = 1 with 304 = floor((2^3+0^3)/3) + floor((10^3+8^3)/5).
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MATHEMATICA
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CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
tab={}; Do[r=0; Do[If[CQ[5(n-Floor[(x^3+y^3)/3])+s-z^3], r=r+1], {s, 0, 4}, {x, 1, (3n+2)^(1/3)}, {y, 0, Min[x-1, (3n+2-x^3)^(1/3)]}, {z, 0, ((5(n-Floor[(x^3+y^3)/3])+s)/2)^(1/3)}]; tab=Append[tab, r], {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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