|
%I #12 Dec 14 2017 10:23:10
%S 1,1,1,1,3,3,1,1,4,3,2,1,3,4,1,1,4,4,2,3,5,4,1,3,5,5,3,1,6,6,1,1,5,5,
%T 3,4,4,6,1,3,8,4,2,2,9,6,1,1,6,8,4,3,6,10,3,4,6,4,5,1,7,7,2,1,9,8,2,5,
%U 9,10
%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x > 0, y >= 0 and 0 <= z <= w such that 64*x^2 + 65*y^2 = q^2 for some practical number q.
%C Conjecture: (i) a(n) > 0 for all n > 0. Also, a(n) = 1 only for those numbers 2^k (k = 0,1,2,...), 4^k*m (k = 0,1,...; m = 3, 7), 4^k*47 (k = 0,1,2,3), 4^k*s (k = 0,1,2; s = 15, 23, 31, 39, 71); 4^k*t (k =0,1; t = 87, 111, 119, 159, 191, 311).
%C (ii) Any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that a*x^2 + b*y^2 = q^2 for some practical number q, provided that (a,b) is among the ordered pairs (5,16), (7,36), (16,77), (36,55), (36,91), (36, -5), (64,-7).
%H Zhi-Wei Sun, <a href="/A296523/b296523.txt">Table of n, a(n) for n = 1..10000</a>
%H 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.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/176r.pdf">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>.)
%e a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 and 64*1^2 + 65*0^2 = 8^2 with 8 practical.
%e a(71) = 1 since 71 = 3^2 + 6^2 + 1^2 + 5^2 and 64*3^2 + 65*6^2 = 54^2 with 54 practical.
%e a(159) = 1 since 159 = 5^2 + 10^2 + 3^2 + 5^2 and 64*5^2 + 65*10^2 = 90^2 with 90 practical.
%e a(311) = 1 since 311 = 1^2 + 2^2 + 9^2 + 15^2 and 64*1^2 + 65*2^2 = 18^2 with 18 practical.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t f[n_]:=f[n]=FactorInteger[n];
%t Pow[n_, i_]:=Pow[n,i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);
%t Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}];
%t pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);
%t pQ[n_]:=pQ[n]=SQ[n]&&pr[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&pQ[64x^2+65y^2],r=r+1],{x,1,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[(n-x^2-y^2)/2]}];tab=Append[tab,r],{n,1,70}]
%Y Cf. A000118, A000290, A271518, A281976.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Dec 14 2017
|
