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A332664 a(n) = number of nonnegative integers that are not the sum of {2 squares, a nonnegative 5th power, and a nonnegative n-th power}.

%I A332664 #19 Mar 20 2020 20:02:02
%S A332664 0,2,14,115,116,109,245,381,1387,913,1234,1552,2103,2838,3036,3384,
%T A332664 4693,5405,8304,9088,11089,13289,15815,18619,20979,22755,24107,24984,
%U A332664 25548
%N A332664 a(n) = number of nonnegative integers that are not the sum of {2 squares, a nonnegative 5th power, and a nonnegative n-th power}.
%C A332664 a(2) = 0 by a theorem of Zhi-Wei Sun, see A273915. All terms beyond a(2) are conjectures and have only been checked to 4*10^9.
%H A332664 W. Jagy and I. Kaplansky, <a href="https://projecteuclid.org/euclid.em/1062621075">Sums of Squares, Cubes and Higher Powers</a>, Experimental Mathematics, vol. 4 (1995) pp. 169-173.
%e A332664 a(2) = 0, since any nonnegative integer k is the sum of 3 squares and a nonnegative 5th power (see A273915).
%e A332664 a(4) = 14. Since any nonnegative integer k (<= 4*10^9) is the sum of {2 squares, a nonnegative 5th power, and a 4th power}, except for 14 numbers: 23, 44, 71, 79, 215, 383, 863, 1439, 1583, 1727, 1759, 1919, 2159, 2543.
%t A332664 a(5)
%t A332664 Do[m=1000000 (k-1)+1; n=1000000 k;
%t A332664   t=Union@Flatten@Table[x^2 + y^2 + z^5 + w^5,
%t A332664 {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/5)},
%t A332664 {w, If[x^2 + y^2 + z^5 < m, Floor[(m-1-x^2-y^2-z^5)^(1/5)] + 1, z], (n-x^2-y^2-z^5)^(1/5)}];
%t A332664   b=Complement[Range[m, n], t];
%t A332664   Print[Length@b], {k,4000}]
%Y A332664 Cf. A022566, A111151, A273915, A319052, A322122.
%K A332664 nonn,more
%O A332664 2,2
%A A332664 _XU Pingya_, Feb 18 2020