A280991 Positive integers that can be expressed as the sum of four pairwise coprime squares.

3, 4, 7, 12, 15, 19, 27, 28, 31, 36, 39, 43, 51, 52, 55, 60, 63, 67, 75, 76, 79, 84, 87, 91, 99, 103, 108, 111, 115, 123, 124, 127, 132, 135, 139, 147, 148, 151, 156, 159, 163, 171, 172, 175, 180, 183, 187, 195, 196, 199, 204, 207, 211, 219, 220, 223, 228, 231, 235, 243, 244, 247

Offset: 1

Author

Emmanuel Vantieghem, Jan 12 2017

Keywords

• nonn

Comments

If n is in the sequence, then n == 0 or 1 mod 3 and n == 3, 4, or 7 mod 8. But the converse is not true: 100 and 268 are not in the sequence (are there other examples?).

Guy [op. cit.] quotes Paul Turan as asking for a characterization of the terms of this sequence. - N. J. A. Sloane, Jan 16 2017

Examples

			3 is in the sequence, since 3 is the sum of the squares of 0, 1, 1, 1 and these four numbers are pairwise coprime.
7 is in the sequence, since 7 is the sum of the squares of 1, 1, 1, 2 and these four numbers are pairwise coprime.
		

References

• R. K. Guy, Unsolved Problems in Theory of Numbers, Section C20

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..1000

Programs

Mathematica

f[A_]:=Module[{A2, La2},A2=Subsets[A,{2}];La2=Length[A2];Union[Table[GCD@@A2[[i]],{i,1,La2}]]=={1}];
Select[Range[250],MemberQ[Union[f/@PowersRepresentations[#,4,2]],True]&]