A335558 Positive integers that cannot be expressed as the sum of at most 5 pairwise coprime squares.
21, 22, 23, 24, 33, 45, 46, 47, 48, 57, 69, 70, 71, 72, 81, 93, 94, 95, 96, 105, 117, 118, 119, 120, 129, 141, 142, 143, 144, 153, 154, 161, 165, 166, 167, 168, 177, 189, 190, 191, 192, 201, 209, 213, 214, 215, 216, 217, 225, 237, 238, 239, 240, 246, 249, 261
Offset: 1
Keywords
References
- R. K. Guy, Unsolved Problems in Number Theory, C20.
Programs
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Mathematica
n = 261; a1 = Prime[Range[6]]^2; a2 = a3 = a4 = a5 = {}; Do[If[GCD[x, y] == 1, AppendTo[a2, x^2 + y^2]], {x, 0, (n/2)^(1/2)}, {y, x, (n - x^2)^(1/2)}]; Do[If[GCD[x, y] == GCD[x, z] == GCD[y, z] == 1, AppendTo[a3, x^2 + y^2 + z^2]], {x, 0, (n/3)^(1/2)}, {y, x, ((n - x^2)/2)^(1/2)}, {z, y, (n - x^2 - y^2)^(1/2)}]; Do[If[GCD[x, y] == GCD[x, z] == GCD[x, t] == GCD[y, z] == GCD[y, t] == GCD[z, t] == 1, AppendTo[a4, x^2 + y^2 + z^2 + t^2]], {x, 0, (n/4)^(1/2)}, {y, x, ((n - x^2)/3)^(1/2)}, {z, y, ((n - x^2 - y^2)/2)^(1/2)}, {t, z, (n - x^2 - y^2 - z^2)^(1/2)}]; Do[If[GCD[x, y] == GCD[x, z] == GCD[x, t] == GCD[x, w] == GCD[y, z] == GCD[y, t] == GCD[y, w] == GCD[z, t] == GCD[z, w] == GCD[t, w] == 1, AppendTo[a5, x^2 + y^2 + z^2 + t^2 + w^2]], {x, 0, (n/5)^(1/2)}, {y, x, ((n - x^2)/4)^(1/2)}, {z, y, ((n - x^2 - y^2)/3)^(1/2)}, {t, z, ((n - x^2 - y^2 - z^2)/2)^(1/2)}, {w, t, (n - x^2 - y^2 - z^2 - t^2)^(1/2)}]; Complement[Range[n], Union@Join[a1, a2, a3, a4, a5]]