cp's OEIS Frontend

A267861 Number of ways to write n as 2*t + u^4 + v^4 + 2*w^4 + 3*x^4 + 4*y^4 + 6*z^4, where t is 0 or 1, and u, v, w, x, y, z are nonnegative integers with u <= v and v > 0.

%I A267861 #27 Jan 08 2025 11:01:17
%S A267861 1,1,2,3,3,4,5,5,5,6,5,5,5,4,3,4,3,3,4,3,4,5,5,5,6,5,5,5,4,3,3,3,2,4,
%T A267861 2,4,4,5,5,6,5,5,6,4,4,3,3,2,4,2,4,4,4,5,6,5,6,6,4,4,4,3,2,4,2,4,5,6,
%U A267861 5,8
%N A267861 Number of ways to write n as 2*t + u^4 + v^4 + 2*w^4 + 3*x^4 + 4*y^4 + 6*z^4, where t is 0 or 1,  and u, v, w, x, y, z are nonnegative integers with u <= v and v > 0.
%C A267861 Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 1, 2, 111, 127, 143, 158, 221, 223, 240, 460, 463, 480, 545, 560, 561, 1455, 1695, 1776, 2175. Moreover, any integer n > 10^4 not among 10543, 17935, 37583, 40383, 78543 can be written as u^4 + v^4 + 2*w^4 + 3*x^4 + 4*y^4 + 6*z^4 with u,v,w,x,y,z nonnegative integers.
%C A267861 If a(1),...,a(7) are positive integers with a(1) <= a(2) <= ... <= a(7) and a(1)+...+a(7) = g(4) = 19 such that {a(1)*x(1)^4+...+a(7)*x(7)^4: x(1),...,x(7) = 0,1,2,...} = {0,1,2,...}, then the tuple (a(1),...,a(7)) must be (1,1,2,2,3,4,6) or (1,1,2,2,3,3,7). Similarly, if a(1),...,a(8) are positive integers with a(1) <= a(2) <= ... <= a(8) and a(1)+...+a(8) = g(5) = 37 such that {a(1)*x(1)^5+...+a(8)*x(8)^5: x(1),...,x(8) = 0,1,2,...} = {0,1,2,...}, then (a(1),...,a(8)) must be (1,1,2,3,4,6,8,12) or (1,1,2,3,4,5,7,14).
%H A267861 Zhi-Wei Sun, <a href="/A267861/b267861.txt">Table of n, a(n) for n = 1..10000</a>
%H A267861 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;3656d479.1604">Upgrade Waring's Problem</a>, a message to Number Theory Mailing List, April 2, 2016.
%H A267861 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.
%e A267861 a(111) = 1 since 111 = 2*1 + 2^4 + 3^4 + 2*1^4 + 3*0^4 + 4*1^4 + 6*1^4.
%e A267861 a(240) = 1 since 240 = 2*0 + 2^4 + 2^4 + 2*0^4 + 3*2^4 + 4*2^4 + 6*2^4.
%e A267861 a(1776) = 1 since 1776 = 2*0 + 4^4 + 5^4 + 2*3^4 + 3*3^4 + 4*1^4 + 6*3^4.
%e A267861 a(2175) = 1 since 2175 = 2*1 + 0^4 + 4^4 + 2*2^4 + 3*5^4 + 4*1^4 + 6*1^4.
%t A267861 QQ[n_]:=QQ[n]=n>0&&IntegerQ[n^(1/4)]
%t A267861 Do[r=0;Do[If[QQ[n-2t-6*z^4-4y^4-3x^4-2w^4-u^4],r=r+1],{t,0,Min[1,n/2]},{z,0,((n-2t^8)/6)^(1/4)},{y,0,((n-2t-6z^4)/4)^(1/4)},{x,0,((n-2t-6z^4-4y^4)/3)^(1/4)},
%t A267861 {w,0,((n-2t-6z^4-4y^4-3x^4)/2)^(1/4)},{u,0,((n-2t-6z^4-4y^4-3x^4-2w^4)/2)^(1/4)}];Print[n," ",r];Continue,{n,1,70}]
%Y A267861 Cf. A000583, A000584, A002804, A267826, A271099, A271169, A271237.
%K A267861 nonn
%O A267861 1,3
%A A267861 _Zhi-Wei Sun_, Apr 07 2016