A289740 Prime powers P for which the number of modulo P residues among sums of three sixth powers is less than P.

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

Author

Jon E. Schoenfield, Jul 10 2017

Keywords

• nonn

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.

Examples

			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.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..56 (based on b-file for A289631 from Giovanni Resta)

Crossrefs

Subsequence of A289631 (similar sequence for sums of two sixth powers).

Cf. A289760 (similar sequence for sums of four sixth powers).

Extensions

a(40)-a(46) added (based on b-file for A289631 from Giovanni Resta) by Jon E. Schoenfield, Jul 15 2017