A289760 -id:A289760 - cp's OEIS frontend

A289631 Prime powers P for which the number of modulo P residues among sums of two sixth powers is less than P.

4, 7, 8, 9, 13, 16, 19, 27, 31, 32, 37, 43, 49, 61, 64, 67, 73, 79, 81, 109, 121, 128, 139, 169, 223, 243, 256, 343, 361, 512, 529, 729, 961, 1024, 1331, 1369, 1849, 2048, 2187, 2197, 2209, 2401, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6561, 6859, 6889, 8192

Offset: 1

Jon E. Schoenfield, Jul 08 2017

nonn

Numbers P in A246655 (prime powers) for which A289630(P) < P.
Every number > 3 that is a power of 2, 3, or 7 is in the sequence.
Primes in this sequence begin 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 109, 139, 223.
Conjecture: 223 is the final prime in this sequence.
From Jon E. Schoenfield, Jul 14 2017: (Start)
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.
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.
(End)

			7 is in the sequence because A289630(7) = 3 < 7.
5 is not in the sequence because A289630(5) = 5.
A289630(12) = 9 < 12, but 12 is not in the sequence because it is not a prime power.

Giovanni Resta, Table of n, a(n) for n = 1..192 (terms < 2*10^6)

Cf. A246655 (prime powers), A289630 (Number of modulo n residues among sums of two sixth powers).

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

isok(n) = isprimepower(n) && (#Set(vector(n^2, i, ((i%n)^6 + (i\n)^6) % n)) < n); \\ Michel Marcus, Jul 11 2017

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

Original entry on oeis.org

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

cp's OEIS Frontend

A289631 Prime powers P for which the number of modulo P residues among sums of two sixth powers is less than P.

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