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

Author

Jonathan Sondow, Jun 28 2017

Keywords

• nonn

Comments

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

Examples

			{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.
		

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

Crossrefs

Cf. A247384, A289118, A289119.

Programs

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

Formula

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

Extensions

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