|
%I #30 Jul 16 2017 02:30:06
%S 7,8,9,13,16,19,27,31,32,49,64,81,128,169,243,256,343,361,512,729,961,
%T 1024,2048,2187,2197,2401,4096,4489,6241,6561,6859,8192,16384,16807,
%U 19321,19683,28561,29791,32768,49729,59049,65536,117649,130321,131072,177147
%N Prime powers P for which the number of modulo P residues among sums of three sixth powers is less than P.
%C Conjecture: the largest prime in the sequence is 31. (If this is true, then the next terms after 32768 are 49729, 59049, and 65536.)
%C Every number > 4 that is a power of 2, 3, or 7 is in the sequence.
%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.
%H Jon E. Schoenfield, <a href="/A289740/b289740.txt">Table of n, a(n) for n = 1..56 (based on b-file for A289631 from Giovanni Resta)</a>
%e 5 is not in the sequence because (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 (j^6 + k^6 + m^6) mod 7 can take on only 4 values (0..3), not 7.
%e 14 is not in the sequence because -- although (j^6 + k^6 + m^6) mod 14 can take on only the 8 (not 14) values 0, 1, 2, 3, 7, 8, 9, and 10 -- 14 is not a prime power.
%Y Subsequence of A289631 (similar sequence for sums of two sixth powers).
%Y Cf. A289760 (similar sequence for sums of four sixth powers).
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Jul 10 2017
%E a(40)-a(46) added (based on b-file for A289631 from _Giovanni Resta_) by _Jon E. Schoenfield_, Jul 15 2017
|
