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A303637
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Number of ways to write n as x^2 + y^2 + 2^z + 5*2^w, where x,y,z,w are nonnegative integers with x <= y.
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21
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0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 4, 3, 4, 5, 5, 5, 4, 4, 5, 4, 5, 9, 8, 6, 6, 9, 7, 6, 8, 8, 10, 8, 4, 8, 5, 7, 9, 12, 9, 6, 10, 9, 11, 10, 8, 16, 12, 8, 9, 12, 9, 11, 12, 11, 9, 10, 12, 14, 10, 10
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OFFSET
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1,7
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COMMENTS
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Conjecture: a(n) > 0 for all n > 5.
This has been verified for n up to 5*10^9. Note that 321256731 cannot be written as x^2 + (2*y)^2 + 2^z + 5*2^w with x,y,z,w nonnegative integers.
In contrast, Crocker proved in 2008 that there are infinitely many positive integers not representable as the sum of two squares and at most two powers of 2.
570143 cannot be written as x^2 + y^2 + 2^z + 3*2^w with x,y,z,w nonnegative integers, and 2284095 cannot be written as x^2 + y^2 + 2^z + 7*2^w with x,y,z,w nonnegative integers.
Jiao-Min Lin (a student at Nanjing University) found a counterexample to the conjecture: a(18836421387) = 0. - Zhi-Wei Sun, Jul 21 2022
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REFERENCES
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R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.
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LINKS
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EXAMPLE
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a(6) = 1 with 6 = 0^2 + 0^2 + 2^0 + 5*2^0.
a(8) = 2 with 8 = 1^2 + 1^2 + 2^0 + 5*2^0 = 0^2 + 1^2 + 2^1 + 5*2^0.
a(9) = 2 with 9 = 1^2 + 1^2 + 2^1 + 5*2^0 = 0^2 + 0^2 + 2^2 + 5*2^0.
a(10) = 2 with 10 = 0^2 + 2^2 + 2^0 + 5*2^0 = 0^2 + 1^2 + 2^2 + 5*2^0.
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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-5*2^k-2^m], Do[If[SQ[n-5*2^k-2^m-x^2], r=r+1], {x, 0, Sqrt[(n-5*2^k-2^m)/2]}]], {k, 0, Log[2, n/5]}, {m, 0, If[n/5==2^k, -1, Log[2, n-5*2^k]]}]; tab=Append[tab, r], {n, 1, 60}]; Print[tab]
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CROSSREFS
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Cf. A000079, A000290, A001481, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, 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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