A319052 - OEIS

%I #18 Jun 30 2019 10:40:33

%S 23,71,120,312,455,2136,99295

%N Numbers that are not the sum of {2 squares, a nonnegative cube, and a nonnegative k-th power with k >= 17}.

%C Subsequence of A022552.

%C Except for the above 7 terms, the remaining 427 numbers in A022552 can be expressed as the sum of two squares, a nonnegative cube and a nonnegative k-th power. So a(n) has only 7 terms, until n = 10^10.

%C Also, for n <= 6*10^7, when k = 3, the number of such forms is only 23; when 4 <= k <= 5, only 23 and 71; when k = 6, only 23, 71 and 455; when 7 <= k <= 8, only 23, 71 and 120; when 9 <= k <= 11, only 23, 71, 120, 312 and 455; when 12 <= k <= 16, only 23, 71, 120, 312, 455 and 2136.

%H 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.

%t n=100000;

%t t=Union@Flatten@Table[x^2+y^2+z^3+w^17, {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/3)}, {w,0,(n-x^2-y^2-z^3)^(1/17)}];

%t Complement[Range[0,n], t]

%Y Cf. A022552, A297930.

%K nonn,more

%O 1,1

%A _XU Pingya_, Sep 08 2018