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A289760 Prime powers P for which the number of modulo P residues among sums of four sixth powers is less than P. 2

%I

%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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Last modified October 25 23:28 EDT 2021. Contains 348256 sequences. (Running on oeis4.)