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A343862 Number of ways to write n as x + y + z with x, y, z positive integers such that x^2*y^2 + 5*y^2*z^2 + 10*z^2*x^2 is a square.

%I A343862 #31 Mar 11 2023 06:56:05
%S A343862 0,0,1,1,1,2,2,1,1,4,2,4,2,2,5,5,3,3,2,6,4,3,3,6,6,5,2,6,4,10,3,6,4,6,
%T A343862 6,8,5,6,4,9,7,6,3,7,9,5,5,15,5,12,11,10,5,6,7,10,8,9,7,15,7,6,7,10,
%U A343862 10,7,9,10,10,12,4,15,9,9,11,9,7,12,11,15,8,9,7,12,10,3,9,11,11,19,12,12,9,10,6,23,11,6,10,18
%N A343862 Number of ways to write n as x + y + z with x, y, z positive integers such that x^2*y^2 + 5*y^2*z^2 + 10*z^2*x^2 is a square.
%C A343862 Conjecture 1: a(n) > 0 for all n > 2.
%C A343862 We have verified a(n) > 0 for all n = 3..10000. Conjecture 1 holds if a(p) > 0 for each odd prime p. For any n > 0 we have a(3*n) > 0 since 3*n = n + n + n and 1 + 5 + 10 = 4^2.
%C A343862 See also A340274 for a similar conjecture.
%C A343862 Conjecture 2: There are infinitely many triples (a,b,c) of positive integers such that each n = 3,4,... can be written as x + y + z with x,y,z positive integers and a*x^2*y^2 + b*y^2*z^2 + c*z^2*x^2 a square.
%C A343862 Such triple candidates include (21,19,9), (23,17,9), (24,16,9), (25,14,10), (29,19,16), (33,27,21), (35,9,5), (37,32,31) etc.
%C A343862 Conjecture 1 holds for all n < 2^15. - _Martin Ehrenstein_, May 02 2021
%H A343862 Martin Ehrenstein, <a href="/A343862/b343862.txt">Table of n, a(n) for n = 1..32767</a> (first 1500 terms from Zhi-Wei Sun)
%H A343862 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190. See also <a href="http://arxiv.org/abs/1604.06723">arXiv:1604.06723 [math.NT]</a>.
%e A343862 a(4) = 1 with 4 = 2 + 1 + 1 and 2^2*1^2 + 5*1^2*1^2 + 10*1^2*2^2 = 7^2.
%e A343862 a(5) = 1 with 5 = 1 + 3 + 1 and 1^2*3^2 + 5*3^2*1^2 + 10*1^1*1^2 = 8^2.
%e A343862 a(8) = 1 with 8 = 4 + 2 + 2 and 4^2*2^2 + 5*2^2*2^2 + 10*2^2*4^2 = 28^2.
%e A343862 a(9) = 1 with 9 = 3 + 3 + 3 and 3^2*3^2 + 5*3^2*3^2 + 10*3^2*3^2 = 36^2.
%e A343862 a(19) = 2. We have 19 = 4 + 5 + 10 with 4^2*5^2 + 5*5^2*10^2 + 10*10^2*4^2 = 170^2, and 19 = 4 + 13 + 2 with 4^2*13^2 + 5*13^2*2^2 + 10*2^2*4^2 = 82^2.
%t A343862 SQ[n_]:=IntegerQ[Sqrt[n]];
%t A343862 tab={};Do[r=0;Do[If[SQ[x^2*y^2+(n-x-y)^2*(5*y^2+10*x^2)],r=r+1],{x,1,n-2},{y,1,n-1-x}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A343862 Cf. A000041, A000290, A230121, A230747, A231168, A262357, A271719, A273278, A340274.
%K A343862 nonn
%O A343862 1,6
%A A343862 _Zhi-Wei Sun_, May 02 2021