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

1,1

Comments

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)

Links

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

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A047538 A074231 A310938 * A076680 A235623 A001074

Adjacent sequences: A289628 A289629 A289630 * A289632 A289633 A289634

Keyword

nonn

Author

Jon E. Schoenfield, Jul 08 2017

Status

approved