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A334086
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Positive numbers not of the form 2*x^4 + y*(y+1)/2 + z*(z+1)/2 with x,y,z nonnegative integers
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3
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19, 82, 109, 118, 145, 149, 271, 280, 296, 349, 350, 371, 392, 454, 491, 643, 670, 692, 754, 755, 923, 937, 986, 989, 1021, 1031, 1150, 1189, 1210, 1294, 1346, 1372, 1610, 1682, 1699, 1720, 1819, 1913, 2050, 2065, 2141, 2227, 2479, 2524, 2753, 2996, 3184, 3451, 3590, 3805, 3968, 4129, 4139, 4199, 4261, 4706
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OFFSET
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1,1
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COMMENTS
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Conjecture: The sequence has totally 216 terms as listed in the b-file.
As none of the 216 terms in the b-file is divisible by 3, the conjecture implies that for each nonnegative integer n we can write 3*n as 2*x^4 + y*(y+1)/2 + z*(z+1)/2 and hence 12*n+1 = 8*x^4 + (y+z+1)^2 + (y-z)^2, where x,y,z are integers.
Our computation indicates that after the 216-th term 4592329 there are no other terms below 10^8.
It is known that each n = 0,1,2,... can be written as the sum of two triangular numbers and twice a square.
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LINKS
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EXAMPLE
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a(1) = 19 since 19 is the first nonnegative integer which cannot be written as the sum of two triangular numbers and twice a fourth power.
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MATHEMATICA
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TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={}; Do[Do[If[TQ[n-2x^4-y(y+1)/2], Goto[aa]], {x, 0, (n/2)^(1/4)}, {y, 0, (Sqrt[4(n-2x^4)+1]-1)/2}]; tab=Append[tab, n]; Label[aa], {n, 0, 5000}]; 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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