cp's OEIS Frontend

A350857 Number of ways to write n as 2*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with w or x a square.

%I A350857 #18 Jan 26 2022 12:51:39
%S A350857 1,2,3,3,3,3,2,2,2,2,3,3,2,2,1,1,3,3,5,4,5,3,2,2,1,3,5,4,4,3,1,2,4,4,
%T A350857 6,4,5,5,4,2,3,6,5,5,2,3,2,2,3,3,7,4,7,6,3,3,2,3,5,3,1,4,2,2,3,5,6,5,
%U A350857 7,4,3,2,2,4,5,3,3,3,1,1,2,6,8,9,5,7,5,3,4,3,6,6,4,3,2,0,3,6,7,5,7
%N A350857 Number of ways to write n as 2*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with w or x a square.
%C A350857 Conjecture 1: a(n) > 0 except for n = 95, 255.
%C A350857 This has been verified for n up to 10^7.
%C A350857 It seems that a(n) = 1 only for n = 0, 14, 15, 24, 30, 60, 78, 79, 111, 159, 174, 249, 270, 303, 318, 334, 382, 399, 414, 462, 472, 831, 894, 975, 1080, 1151, 1164, 1919, 4080, 6456, 20319.
%C A350857 Conjecture 2: For any positive integer a == 2 (mod 4), all sufficiently large integers can be written as a*w^4 + x^4 + (2*y)^2 + z^2 with w,x,y,z integers. If N(a) denotes the largest integer not of the form a*w^4 + x^4 + (2*y)^2 + z^2 (w,x,y,z = 0,1,2,...), then we have N(2) = 255, N(6) = 2716, N(10) = 598, N(14) = 8427, N(18) = 2463, N(22) = 3884, N(26) = 14988, N(30) = 10843.
%C A350857 See also A346643 and A347865 for similar conjectures.
%H A350857 Zhi-Wei Sun, <a href="/A350857/b350857.txt">Table of n, a(n) for n = 0..10000</a>
%H A350857 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.
%H A350857 Zhi-Wei Sun, <a href="http://arxiv.org/abs/2010.05775">Sums of four rational squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020-2022.
%e A350857 a(24) = 1 with 24 = 2*0^4 + 2^4 + 2^2 + 2^2 and 0 = 0^2.
%e A350857 a(1151) = 1 with 1151 = 2*3^4 + 4^4 + 2^2 + 27^2 and 4 = 2^2.
%e A350857 a(6456) = 1 with 6456 = 2*1^4 + 3^4 + 17^2 + 78^2 and 1 = 1^2.
%e A350857 a(20319) = 1 with 20319 = 2*5^4 + 0^4 + 5^2 + 138^2 and 0 = 0^2.
%t A350857 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A350857 tab={};Do[r=0;Do[If[SQ[n-2w^4-x^4-y^2]&&(SQ[w]||SQ[x]),r=r+1],{w,0,(n/2)^(1/4)},{x,0,(n-2w^4)^(1/4)},{y,0,Sqrt[(n-2w^4-x^4)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y A350857 Cf. A000290, A000583, A346643, A347865.
%K A350857 nonn
%O A350857 0,2
%A A350857 _Zhi-Wei Sun_, Jan 19 2022