A247967 a(n) is the smallest k such that prime(k+i) (mod 6) takes successively the values 5, 5, ... for i = 0, 1, ..., n-1.

3, 9, 15, 54, 290, 987, 4530, 21481, 58554, 60967, 136456, 136456, 673393, 1254203, 1254203, 7709873, 21357253, 21357253, 25813464, 25813464, 39500857, 39500857, 947438659, 947438659, 947438659, 5703167678, 5703167678, 16976360924, 68745739764, 117327812949

Offset: 1

Author

Michel Lagneau, Sep 28 2014

Keywords

• nonn

Comments

Weakening the definition to prime(k+i) == 2 (mod 3) yields a(1) = 1, but all other terms are unchanged. See also A247816 (residue 5) or A276414 (equal residues, all 1 or all -1). - M. F. Hasler, Sep 02 2016

Examples

			a(1)= 3 => prime(3) == 5 (mod 6).
a(2)= 9 => prime(9) == 5 (mod 6), prime(10) == 5 (mod 6).
a(3)= 15 => prime(15) == 5 (mod 6), prime(16) == 5 (mod 6), prime(17) == 5 (mod 6).
From _Michel Marcus_, Sep 30 2014: (Start)
The resulting primes are:
  5;
  23, 29;
  47, 53, 59;
  251, 257, 263, 269;
  1889, 1901, 1907, 1913, 1931;
  7793, 7817, 7823, 7829, 7841, 7853;
  43451, 43457, 43481, 43487, 43499, 43517, 43541;
  243161, 243167, 243197, 243203, 243209, 243227, 243233, 243239;
  ... (End)
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..35

Crossrefs

Cf. A057622, A247816, A276414.

Programs

MATLAB

N = 2*10^8; % to use primes up to N
P = mod(primes(N),6);
P5 = find(P==5);
n5 = numel(P5);
a(1) = P5(1);
for k = 2:100
  r = find(P5(k:n5) == P5(1:n5+1-k)+k-1,1,'first');
  if numel(r) == 0
     break
  end
  a(k) = P5(r);
end
a % Robert Israel, Sep 02 2016

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 = 5
        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

Table[k = 1; While[Times @@ Boole@ Map[Mod[Prime[k + #], 6] == 5 &, Range[0, n - 1]] == 0, k++]; k, {n, 10}] (* Michael De Vlieger, Sep 02 2016 *)

PARI

a(n) = {k = 1; ok = 0; while (!ok, nb = 0; for (i=0, n-1, if (prime(k+i) % 6 == 5, nb++, break);); if (nb == n, ok=1, k++);); k;} \\ Michel Marcus, Sep 28 2014

PARI

m=c=i=0;forprime(p=1,, i++;p%6!=5&&(!c||!c=0)&&next; c++>m||next; print1(1+i-m=c,",")) \\ M. F. Hasler, Sep 02 2016

Formula

a(n) = primepi(A057622(n)). - Michel Marcus, Oct 01 2014

Extensions

a(11)-a(22) from A057622 by Michel Marcus, Oct 03 2014

a(23)-a(25) from Jinyuan Wang, Jul 08 2019

a(26)-a(30) added using A057622 by Jinyuan Wang, Apr 15 2020