A289119 Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 6*k+1 and 6*k+5 or 6*k+5 and 6*k+1.

5, 5, 5, 5, 5, 5, 5, 89, 89, 809, 809, 809, 809, 809, 809, 809, 809, 809, 809, 809, 809, 3954889, 15186319, 15186319, 15186319, 77011289, 77011289, 77011289, 288413159, 288413159, 288413159, 288413159, 288413159, 62585146739, 114058236679, 143014298809

Offset: 1

Author

Jonathan Sondow, Jun 25 2017

Keywords

• nonn

Comments

Conjecture: the sequence is infinite. (Motivation: the string HTHTHT... of length n eventually occurs in any sufficiently long sequence of coin tosses.)

Examples

			For k = 3, 4, ..., 33, {Prime[k], Mod[Prime[k], 6]} = {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}, {83, 5}, {89, 5}, {97, 1}, {101, 5}, {103, 1}, {107, 5}, {109, 1}, {113, 5}, {127, 1}, {131, 5}, {137, 5}, so a(n) = 5, 5, 5, 5, 5, 5, 5, 89, 89 for n = 1, 2, ..., 9 with a(10) > 89.
		

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. A057620, A057622, A289118.

Programs

Mathematica

j = 3; T = Table[ While[ Product[ Mod[ Prime[k + 1] - Prime[k], 6], {k, j, j + n}] == 0, j++]; Prime[j], {n, 0, 20}]; Prepend[T, 5]

Extensions

a(23)-a(36) from Giovanni Resta, Jun 29 2017