A273429 - OEIS

%I #18 Feb 11 2017 11:21:59

%S 1,2,2,2,2,2,2,1,1,3,3,2,2,2,2,1,1,3,4,3,2,2,2,1,1,3,4,4,2,2,3,1,1,3,

%T 4,3,3,3,3,2,1,4,4,2,2,3,3,1,1,3,5,5,3,3,5,3,1,3,3,3,2,2,4,2,2,5,7,5,

%U 4,5,4,1,3,6,6,6,4,4,4,1,2

%N Number of ordered ways to write n as x^6 + y^2 + z^2 + w^2, where x,y,z,w are nonnegative integers with y <= z <= w.

%C The author proved in arXiv:1604.06723 that for each c = 1, 4 any natural number can be written as c*x^6 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers. Thus a(n) > 0 for all n = 0,1,2,....

%C We note that a(n) = 1 for the following values of n not divisible by 2^6:  7, 8, 15, 16, 23, 24, 31, 32, 40, 47, 48, 56, 71, 79, 92, 112, 143, 176, 191, 240, 304, 368, 560, 624, 688, 752, 1072, 1136, 1456, 1520, 1840, 1904, 2608, 2672, 3760, 3824, 6512, 6896.

%C For more conjectural refinements of Lagrange's four-square theorem, one may consult the author's preprint arXiv:1604.06723.

%H Zhi-Wei Sun, <a href="/A273429/b273429.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.NT], 2016-2017.

%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.

%e a(7) = 1 since 7 = 1^6 + 1^2 + 1^2 + 2^2 with 1 = 1 < 2.

%e a(8) = 1 since 8 = 0^6 + 0^2 + 2^2 + 2^2 with 0 < 2 = 2.

%e a(15) = 1 since 15 = 1^6 + 1^2 + 2^2 + 3^2 with 1 < 2 < 3.

%e a(16) = 1 since 16 = 0^6 + 0^2 + 0^2 + 4^2 with 0 = 0 < 4.

%e a(56) = 1 since 56 = 0^6 + 2^2 + 4^2 + 6^2 with 2 < 4 < 6.

%e a(71) = 1 since 71 = 1^6 + 3^2 + 5^2 + 6^2 with 3 < 5 < 6.

%e a(79) = 1 since 79 = 1^6 + 2^2 + 5^2 + 7^2 with 2 < 5 < 7.

%e a(92) = 1 since 92 = 1^6 + 1^2 + 3^2 + 9^2 with 1 < 3 < 9.

%e a(143) = 1 since 143 = 1^6 + 5^2 + 6^2 + 9^2 with 5 < 6 < 9.

%e a(191) = 1 since 191 = 1^6 + 3^2 + 9^2 + 10^2 with 3 < 9 < 10.

%e a(624) = 1 since 624 = 2^6 + 4^2 + 12^2 + 20^2 with 4 < 12 < 20.

%e a(2672) = 1 since 2672 = 2^6 + 4^2 + 36^2 + 36^2 with 4 < 36 = 36.

%e a(3760) = 1 since 3760 = 0^6 + 4^2 + 12^2 + 60^2 with 4 < 12 < 60.

%e a(3824) = 1 since 3824 = 2^6 + 4^2 + 12^2 + 60^2 with 4 < 12 < 60.

%e a(6512) = 1 since 6512 = 2^6 + 12^2 + 52^2 + 60^2 with 12 < 52 < 60.

%e a(6896) = 1 since 6896 = 2^6 + 36^2 + 44^2 + 60^2 with 36 < 44 < 60.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

%t Do[r=0;Do[If[SQ[n-x^6-y^2-z^2],r=r+1],{x,0,n^(1/6)},{y,0,Sqrt[(n-x^6)/3]},{z,y,Sqrt[(n-x^6-y^2)/2]}];Print[n," ",r];Continue,{n,0,80}]

%Y Cf. A000118, A000290, A001014, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A270969, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302, A273404, A273432, A273568.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, May 22 2016