A247816 a(n) is the smallest k such that prime(k+i) = 1 (mod 6) for i = 0, 1,...,n-1.

4, 11, 36, 271, 271, 271, 2209, 11199, 13717, 13717, 34369, 172146, 172146, 3094795, 3094795, 4308948, 12762142, 23902561, 72084956, 72084956, 72084956, 1052779161, 1052779161, 1857276773, 1857276773, 19398320447, 57446769091, 57446769091, 57446769091

Offset: 1

Author

Michel Lagneau, Sep 28 2014

Keywords

• nonn
• hard

Comments

Equivalently, "mod 6" can be replaced by "mod 3". See A247967 for the variant "= 5 (mod 6)" and A276414 for runs of primes congruent to each other (mod 3). - M. F. Hasler, Sep 02 2016

The sequence is infinite, by Shiu's theorem. - Jonathan Sondow, Jun 22 2017

Examples

			a(1)= 4 => prime(4) (mod 6)= 1;
a(2)= 11 => prime(11)(mod 6)= 1, prime(12)(mod 6) = 1;
a(3)= 36 => prime(36)(mod 6)= 1, prime(37)(mod 6)= 1, prime(38)(mod 6)= 1.
The resulting primes are:
7;
31, 37;
151, 157, 163;
1741, 1747, 1753, 1759;
1741, 1747, 1753, 1759, 1777;
1741, 1747, 1753, 1759, 1777, 1783;
19471, 19477, 19483, 19489, 19501, 19507, 19531;
... - _Michel Marcus_, Sep 29 2014
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..35
• D. K. L. Shiu, Strings of Congruent Primes, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [MR1760689]

Crossrefs

Cf. A057620, A247967; A276414.

Programs

Maple

for n from 1 to 22 do :
ii:=0:
   for k from 3 to 10^5 while (ii=0)do :
     s:=0:
      for i from 0 to n-1 do:
        r:=irem(ithprime(k+i),6):
        if r = 1
        then
        s:=s+1:
        else
        fi:
      od:
       if s=n and ii=0
       then
       printf ( "%d %d \n",n,k):ii:=1:
       else
       fi:
    od:
od:

Mathematica

With[{m6=If[Mod[#,6]==1,1,0]&/@Prime[Range[5*10^6]]},Flatten[Table[SequencePosition[ m6,PadRight[{},n,1],1],{n,16}],1]][[;;,1]] (* Harvey P. Dale, May 07 2023 *)

PARI

m=c=i=0;forprime(p=1,, i++;p%6!=1&&(!c||!c=0)&&next; c++>m||next; print1(1+i-m=c,",")) \\ M. F. Hasler, Sep 02 2016

Formula

a(n) = primepi(A057620(n)). - Michel Marcus, Sep 30 2014

Extensions

a(12)-a(21) from A057620 by Michel Marcus, Oct 03 2014

a(22)-a(29) from Giovanni Resta, Oct 03 2018