A330938 - OEIS

%I #21 Jan 27 2021 11:37:17

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,22,23,27

%N Numbers that cannot be written as the sum of four proper powers. A proper power is an integer of the form a^b where a,b are integers greater than or equal to 2.

%C There is a proof by Schinzel and Sierpinski that if n >= 33^17 + 12, then n can be written as a sum of four proper powers. Paul Pollack and Enrique Treviño improved that result to find the complete list.

%D Paul Pollack & Enrique Treviño, Sums of Proper Powers, The American Mathematical Monthly, 128 (2021), no. 1, p. 40.

%D A. Schinzel and W. Sierpinski, Sur les puissances propres, Bull. Soc. Roy. Sci. Liege, 34 (1965), pp. 550-554.

%e The first few missing terms are

%e 16 = 2^2 + 2^2 + 2^2 + 2^2,

%e 20 = 2^2 + 2^2 + 2^2 + 2^3,

%e 21 = 2^2 + 2^2 + 2^2 + 3^2,

%e 24 = 2^2 + 2^2 + 2^3 + 2^3,

%e 25 = 2^2 + 2^2 + 2^3 + 3^2,

%e 26 = 2^2 + 2^2 + 3^2 + 3^2,

%e 28 = 2^2 + 2^3 + 2^3 + 2^3.

%Y Cf. A001597.

%K nonn,fini,full

%O 1,2

%A _Enrique Treviño_, Jan 03 2020