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A303429
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Number of ordered pairs (k, m) of nonnegative integers such that n - 3^k - 5^m can be written as the sum of two squares.
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2
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0, 1, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 2, 4, 4, 3, 2, 4, 4, 3, 2, 4, 3, 4, 1, 4, 4, 6, 3, 6, 4, 5, 5, 6, 4, 8, 4, 6, 5, 5, 4, 7, 5, 7, 5, 6, 4, 5, 3, 4, 6, 5, 5, 7, 5, 3, 6, 4, 4, 8, 3, 6, 5, 5, 4, 6, 4, 7, 6, 4, 4, 5, 4, 4, 5, 4, 5, 8, 4, 4, 5, 6, 4, 8, 2, 9, 7, 5, 5, 6
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OFFSET
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1,4
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COMMENTS
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Conjecture: a(n) > 0 for all n > 1.
This is equivalent to the author's conjecture in A303656.
It has been verified that a(n) > 0 for all n = 2..10^9.
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LINKS
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MAPLE
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a(5) = 1 with 5 - 3^1 - 5^0 = 0^2 + 1^2.
a(25) = 1 with 25 - 3^1 - 5^1 = 1^2 + 4^2.
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MATHEMATICA
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SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n], i], 1], 4]==3&&Mod[Part[Part[f[n], i], 2], 2]==1], {i, 1, Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={}; Do[r=0; Do[If[QQ[n-3^k-5^m], r=r+1], {k, 0, Log[3, n]}, {m, 0, If[n==3^k, -1, Log[5, n-3^k]]}]; tab=Append[tab, r], {n, 1, 90}]; Print[tab]
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CROSSREFS
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Cf. A000244, A000290, A000351, A001481, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656.
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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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