A289118 Least prime beginning a string, of length at least n, of consecutive primes which alternate between types 4*k+1 and 4*k+3 or 4*k+3 and 4*k+1.

3, 3, 3, 23, 47, 131, 131, 233, 233, 521, 521, 521, 521, 521, 521, 51749, 505049, 1391087, 2264839, 2556713, 2569529, 2569529, 6160043, 6160043, 6160043, 43679609, 43679609, 198572029, 701575297, 5552898499, 6639843979, 9005520203, 9005520203, 99052377023

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

			{Prime[k], Mod[ Prime[k], 4]} = {{3, 3}, {5, 1}, {7, 3}, {11, 3}, {13, 1}, {17, 1}, {19, 3}, {23, 3}, {29, 1}}, {31, 3}, {37, 1}, . . for k = 2, 3, 4, . ., so a(n) = 3, 3, 3, 23 for n = 1, 2, 3, 4.
		

References

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..45
• Jens Kruse Andersen, Consecutive Congruent Primes

Crossrefs

For the least prime at the start of such a string of length exactly n, see A247384.

Cf. A098058, A289119.

Programs

Mathematica

j = 2; T = Table[ While[ Product[ Mod[ Prime[k + 1] - Prime[k], 4], {k, j, j + n}] == 0,  j++]; Prime[j], {n, 0, 15}]; Prepend[T, 3]

Formula

a(n) = A247384(n) if and only if n > 1 and a(n) < a(n+1).

Extensions

a(18)-a(27) from Alois P. Heinz, Jun 26 2017

a(28)-a(34) from Giovanni Resta, Jul 02 2017