This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A271365 #14 Apr 05 2016 22:16:35
%S A271365 1,4,6,5,5,6,4,1,2,7,9,6,4,3,1,1,6,12,10,4,3,3,1,1,6,12,11,7,6,4,2,4,
%T A271365 9,12,8,5,10,12,6,2,5,9,8,8,10,6,2,3,8,13,10,8,11,8,2,1,6,10,8,7,6,2,
%U A271365 2,7,15,20,14,9,13,11
%N A271365 Number of ordered ways to write n as u^2 + v^3 + x^4 + y^5 + z^6, where u is a positive integer, and v, x, y, z are nonnegative integers.
%C A271365 Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 8, 15, 16, 23, 24, 56. Moreover, the only positive integers not represented by u^2+v^3+x^4+y^5 (u > 0 and v,x,y >= 0) are 8, 15, 23, 55, 62, 71, 471, 478, 510, 646, 806, 839, 879, 939, 1023, 1063, 1287, 2127, 5135, 6811, 7499, 9191, 26471.
%C A271365 Note that 1/2+1/3+1/4+1/5+1/6 = 29/20 < 3/2.
%H A271365 Zhi-Wei Sun, <a href="/A271365/b271365.txt">Table of n, a(n) for n = 1..10000</a>
%H A271365 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2015.10.014">A result similar to Lagrange's theorem</a>, J. Number Theory 162(2016), 190-211.
%e A271365 a(1) = 1 since 1 = 1^2 + 0^3 + 0^4 + 0^5 + 0^6.
%e A271365 a(8) = 1 since 8 = 2^2 + 1^3 + 1^4 + 1^5 + 1^6.
%e A271365 a(15) = 1 since 15 = 2^2 + 2^3 + 1^4 + 1^5 + 1^6.
%e A271365 a(16) = 1 since 16 = 4^2 + 0^3 + 0^4 + 0^5 + 0^6.
%e A271365 a(23) = 1 since 23 = 2^2 + 1^3 + 2^4 + 1^5 + 1^6.
%e A271365 a(24) = 1 since 24 = 4^2 + 2^3 + 0^4 + 0^5 + 0^6.
%e A271365 a(56) = 1 since 56 = 4^2 + 2^3 + 0^4 + 2^5 + 0^6.
%t A271365 SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
%t A271365 Do[r=0;Do[If[SQ[n-x1^6-x2^5-x3^4-x4^3],r=r+1],{x1,0,n^(1/6)},{x2,0,(n-x1^6)^(1/5)},{x3,0,(n-x1^6-x2^5)^(1/4)},{x4,0,(n-x1^6-x2^5-x3^4)^(1/3)}];Print[n," ",r];Continue,{n,1,70}]
%Y A271365 Cf. A000290, A000578, A000583, A000584, A001014, A262813, A262824, A262827, A262857, A266968, A270488, A270516, A270533, A270559, A270566, A270920, A271026, A271106, A271325.
%K A271365 nonn
%O A271365 1,2
%A A271365 _Zhi-Wei Sun_, Apr 05 2016