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%I #28 Jul 16 2017 02:30:21
%S 7,8,9,13,16,27,32,49,64,81,128,169,243,256,343,512,729,961,1024,2048,
%T 2187,2197,2401,4096,6561,8192,16384,16807,19683,28561,29791,32768,
%U 59049,65536,117649,131072,177147,262144,371293,524288,531441,823543,923521,1048576
%N Prime powers P for which the number of modulo P residues among sums of four sixth powers is less than P.
%C If any prime power P = p^k (where p is prime and k >= 1) is in the sequence, then so is p^j for all j > k.
%C The sequence appears to consist of all numbers > 4 that are powers of 2, 3, 7, or 13, and all powers of 31 except 31 itself.
%C It appears that this sequence differs from the similar sequence for sums of five sixth powers only in that that sequence does not contain any powers of 31.
%C Conjecture: the largest prime in the sequence is 13.
%e 5 is not in the sequence because (i^6 + j^6 + k^6 + m^6) mod 5, where j, k, and m are integers, can take on all 5 values 0..4.
%e 7 is in the sequence because (i^6 + j^6 + k^6 + m^6) mod 7 can take on only 5 values (0..4), not 7.
%e 14 is not in the sequence because -- although (i^6 + j^6 + k^6 + m^6) mod 14 can take on only the 10 (not 14) values 0, 1, 2, 3, 4, 7, 8, 9, 10, and 11 -- 14 is not a prime power.
%Y Subsequence of A289740 (similar sequence for sums of three sixth powers).
%Y Cf. A289631 (similar sequence for sums of two sixth powers).
%K nonn,more
%O 1,1
%A _Jon E. Schoenfield_, Jul 11 2017
%E a(30)-a(44) added (using b-file for A289631 from Giovanni Resta) by _Jon E. Schoenfield_, Jul 15 2017
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