A289631 -id:A289631 - cp's OEIS frontend

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

Jon E. Schoenfield, Jul 10 2017

nonn

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.

			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.

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

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

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

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

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

Original entry on oeis.org

7, 8, 9, 13, 16, 27, 32, 49, 64, 81, 128, 169, 243, 256, 343, 512, 729, 961, 1024, 2048, 2187, 2197, 2401, 4096, 6561, 8192, 16384, 16807, 19683, 28561, 29791, 32768, 59049, 65536, 117649, 131072, 177147, 262144, 371293, 524288, 531441, 823543, 923521, 1048576

Offset: 1

Jon E. Schoenfield, Jul 11 2017

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.
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.
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.
Conjecture: the largest prime in the sequence is 13.

			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.
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.
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.

Subsequence of A289740 (similar sequence for sums of three sixth powers).

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

a(30)-a(44) added (using b-file for A289631 from Giovanni Resta) by Jon E. Schoenfield, Jul 15 2017

cp's OEIS Frontend

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

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