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A297995 Number of ways to write n as 4*u + v^2 + x^3 + y^4 + 2*z^8, where u is 0 or 1, v is a positive integer and x,y,z are nonnegative integers.

%I A297995 #12 Jan 18 2018 03:11:36
%S A297995 1,2,2,3,4,4,4,4,5,5,5,5,4,4,4,5,6,5,5,5,5,5,5,5,5,5,6,6,6,6,7,7,6,5,
%T A297995 5,5,7,6,6,5,4,5,5,6,3,5,4,4,5,5,5,5,6,5,5,5,4,4,4,2,2,2,3,4,6,6,5,7,
%U A297995 6,7,5,6,4,4,4,3,4,3,5,4
%N A297995 Number of ways to write n as 4*u + v^2 + x^3 + y^4 + 2*z^8, where u is 0 or 1, v is a positive integer and x,y,z are nonnegative integers.
%C A297995 Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 398, 496, 498, 500, 1507, 6419, 7843, 7983, 8688, 8947, 9175, 9251, 12923, 12976, 48381.
%C A297995 (ii) For any positive integers a,b,c,d,e and integers g,h,i,j,k greater than one, if each n = 0,1,2,... can be written as a*u^g + b*v^h + c*x^i + d*y^j + e*z^k with u,v,x,y,z nonnegative integers, then 1/g + 1/h + 1/i + 1/j + 1/k is greater than 1/2 + 1/3 + 1/4 + 1/8 = 29/24.
%H A297995 Zhi-Wei Sun, <a href="/A297995/b297995.txt">Table of n, a(n) for n = 1..10000</a>
%H A297995 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. (See Remark 3.5.)
%e A297995 a(500) = 1 since 500 = 4*0 + 22^2 + 0^3 + 2^4 + 2*0^8.
%e A297995 a(1507) = 1 since 1507 = 4*1 + 13^2 + 11^3 + 1^4 + 2*1^8.
%e A297995 a(9251) = 1 since 9251 = 4*0 + 91^2 + 7^3 + 5^4 + 2*1^8.
%e A297995 a(12923) = 1 since 12923 = 4*1 + 54^2 + 1^3 + 10^4 + 2*1^8.
%e A297995 a(12976) = 1 since 12976 = 4*0 + 92^2 + 15^3 + 5^4 + 2*2^8.
%e A297995 a(48381) = 1 since 48381 = 4*1 + 140^2 + 6^3 + 13^4 + 2*0^8.
%t A297995 SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
%t A297995 Do[r=0;Do[If[SQ[n-4u-2z^8-y^4-x^3],r=r+1],{u,0,Boole[n>3]},{z,0,((n-4u)/2)^(1/8)},{y,0,(n-4u-2z^8)^(1/4)},{x,0,(n-4u-2z^8-y^4)^(1/3)}];Print[n," ",r],{n,1,80}]
%Y A297995 Cf. A000290, A000578, A000583, A262827, A280356.
%K A297995 nonn
%O A297995 1,2
%A A297995 _Zhi-Wei Sun_, Jan 10 2018