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A338686 Number of ways to write n as x^5 + y^2 + [z^2/7], where x,y,z are integers with x >= 0, y >= 1 and z >= 2, and [.] is the floor function.

%I A338686 #19 Apr 28 2021 04:16:33
%S A338686 1,2,2,3,3,3,3,2,3,4,4,4,3,2,3,4,3,5,5,2,4,3,2,3,4,5,4,4,4,4,2,2,6,5,
%T A338686 2,6,8,6,4,5,6,6,5,4,6,5,4,5,4,10,6,5,8,3,5,5,7,6,4,5,7,5,2,6,7,6,7,8,
%U A338686 6,4,5,6,8,6,2,4,8,4,6,3,5,10,6,8,7,5,5,6,5,5,5,7,6,4,4,6,3,8,7,7
%N A338686 Number of ways to write n as x^5 + y^2 + [z^2/7], where x,y,z are integers with x >= 0, y >= 1 and z >= 2, and [.] is the floor function.
%C A338686 Conjecture: a(n) > 0 for all n > 0.
%C A338686 We have verified a(n) > 0 for all n = 1..7*10^6.
%C A338686 See also A338687 for a similar conjecture.
%C A338686 Conjecture verified up to 2*10^9. - _Giovanni Resta_, Apr 28 2021
%H A338686 Zhi-Wei Sun, <a href="/A338686/b338686.txt">Table of n, a(n) for n = 1..10000</a>
%e A338686 a(1) = 1 with 1 = 0^5 + 1^2 + [2^2/7].
%e A338686 a(166) = 1 with 166 = 0^5 + 1^2 + [34^2/7].
%e A338686 a(323) = 1 with 323 = 2^5 + 17^2 + [4^2/7].
%e A338686 a(815) = 1 with 815 = 2^5 + 1^2 + [74^2/7].
%e A338686 a(2069) = 1 with 2069 = 0^5 + 37^2 + [70^2/7].
%e A338686 a(7560) = 1 with 7560 = 2^5 + 64^2 + [155^2/7].
%e A338686 a(24195) = 1 with 24195 = 0^5 + 8^2 + [411^2/7].
%e A338686 a(90886) = 2 with 90886 = 4^5 + 34^2 + [788^2/7] = 9^5 + 139^2 + [296^2/7].
%t A338686 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A338686 tab={};Do[r=0;Do[If[SQ[n-x^5-Floor[y^2/7]],r=r+1],{x,0,(n-1)^(1/5)},{y,2,Sqrt[7(n-x^5)-1]}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A338686 Cf. A000290, A000584, A270920, A338687, A343387, A343391, A343397.
%K A338686 nonn
%O A338686 1,2
%A A338686 _Zhi-Wei Sun_, Apr 23 2021