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%I #22 Dec 30 2017 09:43:21
%S 1,1,2,2,3,4,4,6,5,7,6,7,7,6,8,6,8,6,7,7,6,8,6,8,6,7,7,6,7,5,6,4,5,4,
%T 3,4,3,4,3,4,4,4,5,4,5,4,5,5,4,5,4,5,4,5,5,4,5,4,5,4,4,4,3,3,4,3,3,3,
%U 4,5,3,6,4,7,5,5,7,4,8,4,7
%N Number of ordered ways to write n as s^5 + t^5 + 2*u^5 + 3*v^5 + 4*w^5 + 5*x^5 + 7*y^5 + 14*z^5, where s,t,u,v,w,x,y,z are nonnegative integers with s <= t.
%C Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 2602.
%C Note that 1+1+2+3+4+5+7+14 = 37. In 1964 J.-R. Chen proved that any natural number can be written as the sum of 37 fifth powers of nonnegative integers.
%C For k = 2,3,4,... define s(k) as the smallest positive integer s such that {a(1)*x(1)^k+...+a(s)*x(s)^k: x(1),...,x(s) = 0,1,2,...} = {0,1,2,...} for some positive integers a(1), ..., a(s), and t(k) as the least positive integer t such that {a(1)*x(1)^k+...+a(t)*x(t)^k: x(1),...,x(t) = 0,1,2,...} = {0,1,2,...} for some positive integers a(1), ..., a(t) with a(1)+...+a(t) = g(k), where g(.) is given by A002804. Then s(k) <= t(k) <= g(k). Part (iii) of the conjecture in A271099 implies that t(k) <= 2k-1 for k > 2. It is easy to see that s(2) = t(2) = 4. Our computation suggests that s(3) = t(3) = 5, s(4) = t(4) = 7, s(5) = t(5) = 8 (which is smaller than 2*5-1), and s(6) = t(6) = 10. We conjecture that s(k) = t(k) for any integer k > 1, and that each natural number can be written as x(1)^6+x(2)^6+x(3)^6+2*x(4)^6+3*x(5)^6+5*x(6)^6+6*x(7)^6+10*x(8)^6+18*x(9)^6+26*x(10)^6, where x(1),x(2),...,x(10) are nonnegative integers. Note that 1+1+1+2+3+5+6+10+18+26 = 73 = g(6).
%C We also conjecture that any natural number can be written as s^5+t^5+2*u^5+3*v^5+4*w^5+6*x^5+8*y^5+12*z^5, with s,t,u,v,w,x,y,z nonnegative integers. Note that 1+1+2+3+4+6+8+12 = 37 = g(5). - _Zhi-Wei Sun_, Apr 04 2016
%D J.-R. Chen, Waring's Problem for g(5)=37, Sci. Sinica 13(1964), 1547-1568.
%H Zhi-Wei Sun, <a href="/A271169/b271169.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 a(1) = 1 since 1 = 0^5 + 1^5 + 2*0^5 + 3*0^5 + 4*0^5 + 5*0^5 + 7*0^5 + 14*0^5.
%e a(2602) = 1 since 2602 = 0^5 + 1^5 + 2*4^5 + 3*2^5 + 4*1^5 + 5*1^5 + 7*0^5 + 14*2^5.
%t FQ[n_]:=FQ[n]=IntegerQ[n^(1/5)]
%t Do[r=0;Do[If[FQ[n-14z^5-7y^5-5x^5-4w^5-3v^5-2u^5-s^5],r=r+1],{z,0,(n/14)^(1/5)},{y,0,((n-14z^5)/7)^(1/5)},{x,0,((n-14z^5-7y^5)/5)^(1/5)},{w,0,((n-14z^5-7y^5-5x^5)/4)^(1/5)},{v,0,((n-14z^5-7y^5-5x^5-4w^5)/3)^(1/5)},{u,0,((n-14z^5-7y^5-5x^5-4w^5-3v^5)/2)^(1/5)}, {s,0,((n-14z^5-7y^5-5x^5-4w^5-3v^5-2u^5)/2)^(1/5)}];Print[n," ",r];Label[aa];Continue,{n,0,80}]
%Y Cf. A000578, A000583, A000584, A001014, A002804, A271099.
%K nonn
%O 0,3
%A _Zhi-Wei Sun_, Mar 31 2016
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