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A308662
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Number of ways to write n as (2^a*5^b)^2 + c*(3c+1) + d*(3d+2), where a and b are nonnegative integers, and c and d are integers.
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3
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1, 1, 1, 2, 2, 3, 1, 2, 3, 1, 3, 2, 2, 2, 2, 4, 2, 2, 5, 3, 3, 3, 3, 3, 4, 6, 4, 3, 3, 5, 4, 4, 3, 6, 5, 6, 3, 2, 6, 3, 6, 2, 3, 4, 4, 6, 5, 5, 4, 4, 6, 1, 4, 4, 4, 6, 3, 5, 2, 6, 7, 3, 2, 5, 5, 4, 5, 6, 8, 5, 6, 5, 4, 8, 3, 7, 3, 3, 7, 3, 6, 7, 4, 4, 7, 7, 4, 4, 8, 7, 4, 3, 6, 4, 7, 7, 4, 1, 6, 7
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OFFSET
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1,4
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COMMENTS
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Conjecture 1: a(n) > 0 for all n > 0.
Conjecture 2: Let r be 1 or 2. Then, any positive integer n can be written as (2^a*5^b)^2 + c*(2c+1) + d*(3d+r), where a and b are nonnegative integers, and c and d are integers.
We have verified Conjectures 1 and 2 for all n = 1..10^8.
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LINKS
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EXAMPLE
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a(3) = 1 with 3 = (2^0*5^0)^2 + (-1)*(3*(-1)+1) + 0*(3*0+2).
a(7) = 1 with 7 = (2^1*5^0)^2 + (-1)*(3*(-1)+1) + (-1)*(3*(-1)+2).
a(10) = 1 with 10 = (2^0*5^0)^2 + 1*(3*1+1) + 1*(3*1+2).
a(52) = 1 with 52 = (2^0*5^0)^2 + 3*(3*3+1) + (-3)*(3*(-3)+2).
a(98) = 1 with 98 = (2^0*5^1)^2 + 4*(3*4+1) + (-3)*(3*(-3)+2).
a(14596) = 1 with 14596 = (2^3*5^0)^2 + (-36)*(3*(-36)+1) + (-60)*(3*(-60)+2).
a(22163) = 1 with 22163 = (2^3*5^0)^2 + 66*(3*66+1) + (-55)*(3*(-55)+2).
a(150689) = 1 with 150689 = (2^6*5^1)^2 + 117*(3*117+1) + (-49)*(3*(-49)+2).
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MATHEMATICA
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OctQ[n_]:=OctQ[n]=IntegerQ[Sqrt[3n+1]];
tab={}; Do[r=0; Do[If[OctQ[n-4^a*25^b-x(3x+1)], r=r+1], {a, 0, Log[4, n]}, {b, 0, Log[25, n/4^a]}, {x, -Floor[(Sqrt[12(n-4^a*25^b)+1]+1)/6], (Sqrt[12(n-4^a*25^b)+1]-1)/6}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
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CROSSREFS
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Cf. A000079, A000351, A001082, A001318, A308566, A308584, A308621, A308623, A308640, A308641, A308644, A308656, A308661.
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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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