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%I #26 Jul 15 2017 11:35:18
%S 4,7,8,9,13,16,19,27,31,32,37,43,49,61,64,67,73,79,81,109,121,128,139,
%T 169,223,243,256,343,361,512,529,729,961,1024,1331,1369,1849,2048,
%U 2187,2197,2209,2401,3481,3721,4096,4489,5041,5329,6241,6561,6859,6889,8192
%N Prime powers P for which the number of modulo P residues among sums of two sixth powers is less than P.
%C Numbers P in A246655 (prime powers) for which A289630(P) < P.
%C Every number > 3 that is a power of 2, 3, or 7 is in the sequence.
%C Primes in this sequence begin 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 109, 139, 223.
%C Conjecture: 223 is the final prime in this sequence.
%C From _Jon E. Schoenfield_, Jul 14 2017: (Start)
%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 Conjecture: the terms in this sequence that are the squares of primes are the squares of 13, 37, 61, 73, 109, and every prime not congruent to 1 mod 4.
%C (End)
%H Giovanni Resta, <a href="/A289631/b289631.txt">Table of n, a(n) for n = 1..192</a> (terms < 2*10^6)
%e 7 is in the sequence because A289630(7) = 3 < 7.
%e 5 is not in the sequence because A289630(5) = 5.
%e A289630(12) = 9 < 12, but 12 is not in the sequence because it is not a prime power.
%o (PARI) isok(n) = isprimepower(n) && (#Set(vector(n^2, i, ((i%n)^6 + (i\n)^6) % n)) < n); \\ _Michel Marcus_, Jul 11 2017
%Y Cf. A246655 (prime powers), A289630 (Number of modulo n residues among sums of two sixth powers).
%Y Cf. A289740 (similar sequence for sums of three sixth powers), A289760 (similar sequence for sums of four sixth powers). - _Jon E. Schoenfield_, Jul 14 2017
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Jul 08 2017
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