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Numbers that are not the sum of nine (9) positive cubes.
5

%I #21 Apr 18 2024 13:49:42

%S 1,2,3,4,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,24,25,26,27,28,

%T 29,31,32,33,34,36,38,39,40,41,43,45,46,47,48,50,52,53,54,55,57,59,60,

%U 62,64,66,67,69,71,73,74,76,78,80,81,83,85,88,90,92,95,97,99,102

%N Numbers that are not the sum of nine (9) positive cubes.

%C The sequence is finite, with last term a(114) = 471.

%H M. F. Hasler, <a href="/A332109/b332109.txt">Table of n, a(n) for n = 1..114</a> (full sequence).

%H Brennan Benfield and Oliver Lippard, <a href="https://arxiv.org/abs/2404.08193">Integers that are not the sum of positive powers</a>, arXiv:2404.08193 [math.NT], 2024. See p. 4.

%e The smallest positive numbers not in the sequence are:

%e 9 = 9 * 1^3, 16 = 2^3 + 8 * 1^3, 23 = 2 * 2^3 + 7 * 1^3,

%e 30 = 3 * 2^3 + 6 * 1^3 and then 35 = 3^3 + 8 * 1^3.

%e The last 10 terms of the sequence are a(105 .. 114) = {293, 305, 310, 312, 319, 347, 366, 373, 422, 471}.

%t Select[Range[500], (pr = PowersRepresentations[#, 9, 3][[;; , 1]]) == {} || Max[pr] == 0 &] (* _Amiram Eldar_, Aug 24 2020 *)

%o (PARI) A332109=setminus([1..555],A003332_upto(666))

%Y Complement of A003332.

%Y Cf. A332107, A332108, A332110 (analog for 7, 8 and 10 cubes, respectively).

%K nonn,fini,full

%O 1,2

%A _M. F. Hasler_, Aug 24 2020