A289237 Find the first (maximal) string, of length exactly n, of consecutive primes that alternate between types 6*k+1 and 6*k+5 or 6*k+5 and 6*k+1. The first element is a(n).

53, 29, 67, 37, 449, 179, 5, 389, 89, 2213, 11149, 10369, 6761, 113341, 80447, 151909, 43777, 2964553, 1457333, 175573, 809, 3954889, 121930481, 96050953, 15186319, 296080717, 98380549, 77011289, 2720227693, 5696814287, 1572386903, 4136299357, 288413159

Offset

1,1

Comments

By the first Formula, a(21) = 809 since A289119(21) = 809 < A289119(22).

References

R. K. Guy, Unsolved Problems in Number Theory, A4.

Links

Giovanni Resta, Table of n, a(n) for n = 1..43

Jens Kruse Andersen, Consecutive Congruent Primes

Formula

a(n) = A289119(n) if and only if n > 1 and A289119(n) < A289119(n+1).

Example

{Prime[k], Mod[Prime[k], 6]} = {2, 2}, {3, 3}, {5, 5}, {7, 1}, {11, 5}, {13, 1}, {17, 5}, {19, 1}, {23, 5}, {29, 5}, {31, 1}, {37, 1}, {41, 5}, {43, 1}, {47, 5}, {53, 5}, {59, 5}, {61, 1}, {67,  1}, {71, 5}, {73, 1}, {79, 1}, . ., so a(n) = 53, 29, 67, 37 for n = 1, 2, 3, 4 and a(7) = 5.

Mathematica

i = 2; While[ Mod[ Prime[i] - Prime[i - 1], 6] != 0 || Mod[ Prime[i + 1] - Prime[i], 6] != 0, i++]; T = {Prime[i]}; Do[j = 3; While[ ! (Product[ Mod[ Prime[k + 1] - Prime[k], 6], {k, j, j + n}] != 0 && (Mod[ Prime[j] - Prime[j - 1], 6] == 0 || j == 3) && Mod[ Prime[j + n + 2] - Prime[j + n + 1], 6] == 0), j++]; T = Append[T, Prime[j]], {n, 0, 16}]; T

Crossrefs

Cf. A247384, A289118, A289119.

Sequence in context: A048602 A048606 A033373 * A319904 A171132 A259091

Adjacent sequences: A289234 A289235 A289236 * A289238 A289239 A289240

Keyword

nonn

Author

Jonathan Sondow, Jun 28 2017

Extensions

a(19)-a(33) from Giovanni Resta, Jun 29 2017

Status

approved