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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. 0
1, 2, 5, 10, 13, 37, 58, 85, 130 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 13 08:18 EDT 2022. Contains 356079 sequences. (Running on oeis4.)