A247967 - OEIS

%I #48 Sep 06 2024 08:06:10

%S 3,9,15,54,290,987,4530,21481,58554,60967,136456,136456,673393,

%T 1254203,1254203,7709873,21357253,21357253,25813464,25813464,39500857,

%U 39500857,947438659,947438659,947438659,5703167678,5703167678,16976360924,68745739764,117327812949

%N 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.

%C 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

%H Amiram Eldar, <a href="/A247967/b247967.txt">Table of n, a(n) for n = 1..35</a>

%F a(n) = primepi(A057622(n)). - _Michel Marcus_, Oct 01 2014

%e a(1)= 3 => prime(3) == 5 (mod 6).

%e a(2)= 9 => prime(9) == 5 (mod 6), prime(10) == 5 (mod 6).

%e a(3)= 15 => prime(15) == 5 (mod 6), prime(16) == 5 (mod 6), prime(17) == 5 (mod 6).

%e From _Michel Marcus_, Sep 30 2014: (Start)

%e The resulting primes are:

%e   5;

%e   23, 29;

%e   47, 53, 59;

%e   251, 257, 263, 269;

%e   1889, 1901, 1907, 1913, 1931;

%e   7793, 7817, 7823, 7829, 7841, 7853;

%e   43451, 43457, 43481, 43487, 43499, 43517, 43541;

%e   243161, 243167, 243197, 243203, 243209, 243227, 243233, 243239;

%e   ... (End)

%p for n from 1 to 22 do :

%p ii:=0:

%p    for k from 3 to 10^5 while (ii=0)do :

%p      s:=0:

%p       for i from 0 to n-1 do:

%p         r:=irem(ithprime(k+i),6):

%p         if r = 5

%p         then

%p         s:=s+1:

%p         else

%p         fi:

%p       od:

%p        if s=n and ii=0

%p        then

%p        printf ( "%d %d \n",n,k):ii:=1:

%p        else

%p        fi:

%p     od:

%p od:

%t 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 *)

%o (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

%o (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

%o (MATLAB)

%o N = 2*10^8; % to use primes up to N

%o P = mod(primes(N),6);

%o P5 = find(P==5);

%o n5 = numel(P5);

%o a(1) = P5(1);

%o for k = 2:100

%o   r = find(P5(k:n5) == P5(1:n5+1-k)+k-1,1,'first');

%o   if numel(r) == 0

%o      break

%o   end

%o   a(k) = P5(r);

%o end

%o a % _Robert Israel_, Sep 02 2016

%Y Cf. A057622, A247816, A276414.

%K nonn

%O 1,1

%A _Michel Lagneau_, Sep 28 2014

%E a(11)-a(22) from A057622 by _Michel Marcus_, Oct 03 2014

%E a(23)-a(25) from _Jinyuan Wang_, Jul 08 2019

%E a(26)-a(30) added using A057622 by _Jinyuan Wang_, Apr 15 2020