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A289740
Prime powers P for which the number of modulo P residues among sums of three sixth powers is less than P.
3
7, 8, 9, 13, 16, 19, 27, 31, 32, 49, 64, 81, 128, 169, 243, 256, 343, 361, 512, 729, 961, 1024, 2048, 2187, 2197, 2401, 4096, 4489, 6241, 6561, 6859, 8192, 16384, 16807, 19321, 19683, 28561, 29791, 32768, 49729, 59049, 65536, 117649, 130321, 131072, 177147
OFFSET
1,1
COMMENTS
Conjecture: the largest prime in the sequence is 31. (If this is true, then the next terms after 32768 are 49729, 59049, and 65536.)
Every number > 4 that is a power of 2, 3, or 7 is in the sequence.
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.
EXAMPLE
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.
7 is in the sequence because (j^6 + k^6 + m^6) mod 7 can take on only 4 values (0..3), not 7.
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.
CROSSREFS
Subsequence of A289631 (similar sequence for sums of two sixth powers).
Cf. A289760 (similar sequence for sums of four sixth powers).
Sequence in context: A297257 A296698 A297131 * A289760 A241289 A037369
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Jul 10 2017
EXTENSIONS
a(40)-a(46) added (based on b-file for A289631 from Giovanni Resta) by Jon E. Schoenfield, Jul 15 2017
STATUS
approved