A140411 Conjectured complete list of squarefree numbers that can be written as a sum of at most two positive squares, but not as a sum of three positive squares.

1, 2, 5, 10, 13, 37, 58, 85, 130

Offset

1,2

Comments

Conjecture 1,9, p. 4, of Goswick et al. "The squarefree numbers in question form a subset of Euler's numeri idonei [A000926], therefore at most one number can be absent from the list above. If such a number does exist, it must exceed 2 * 10^11 and if it is even the Generalized Riemann Hypothesis is false."

Links

Table of n, a(n) for n=1..9.

Lee M. Goswick, Emil W. Kiss, Gabor Moussong, Nandor Simanyi, Sums of squares and orthogonal integral vectors, arXiv:0806.3943 [math.NT], 2011-2013.

Daejun Kim, Jeongwon Lee, Byeong-Kweon Oh, A sum of three nonunit squares of integers, arXiv:1909.04982 [math.NT], 2019.

Formula

a(n) in A005117 and a(n) in {i^2 + j^2 for i,j > 1} and a(n) not in {i^2 + j^2 + k^2 for i,j,k > 1}.

Mathematica

Join[{1}, Select[Range[500], Abs[MoebiusMu[#]] == 1 && Length[Select[PowersRepresentations[#, 2, 2], Not[MemberQ[#, 0, 2]] &]] > 0 && Length[Select[PowersRepresentations[#, 3, 2], Not[MemberQ[#, 0, 2]] &]] == 0 &]] (* Alonso del Arte, Sep 12 2019 *)
Select[Range[500], SquareFreeQ[#] && (p = IntegerPartitions[#, {1, 3}, Range[Sqrt@#]^2]; p != {} && ! MemberQ[Length /@ p, 3]) &] (* Giovanni Resta, Sep 12 2019 *)

Crossrefs

Cf. A000926, A005117.

Sequence in context: A018296 A033316 A099194 * A053353 A099792 A331605

Adjacent sequences: A140408 A140409 A140410 * A140412 A140413 A140414

Keyword

fini,full,nonn

Author

Jonathan Vos Post, Jun 25 2008

Status

approved