A125251 - OEIS

%I #11 Sep 02 2024 01:20:40

%S 5,6,8,9,14,19,43,44,77,85,91,112,113,142,155,195,196,212,226,300,308,

%T 321,351,363,399,456,461,467,485,541,555,602,604,618,638,646,720,728,

%U 779,789,891,896,923,980,1009,1099,1105,1150,1176,1234,1253,1287,1392

%N a(n)=sqrt(A051779(n+2)-1)/30.

%C Consider twin primes p, q = p + 2 such that pq + 2 is prime. It would seem that there are infinitely many such p. Except for p = 3 and p = 5 all such p appear to be of the form 30k - 1 and the values of k give the current sequence. - _James R. Buddenhagen_, Jan 09 2007

%C This is true. Prime numbers (other than 2,3,5) are 30k + 1,7,11,13,17,19,23,29. p+2 is then prime only for p = 30k + 11,17,29; then p(p+2)+2 is 30k + 25,25,1 respectively, so the last case mod 30 is the only one possible. - _Gareth McCaughan_, Jan 09 2007

%C This is the sequence of positive integers k such that p = 30*k - 1, q = 30*k + 1 and p*q + 2 are all prime. - _James R. Buddenhagen_, Jan 09 2007

%H Zak Seidov, <a href="/A125251/b125251.txt">Table of n, a(n) for n=1..2000</a>

%e a(1)=5 because A051779(3)=22501 and sqrt(22501-1)/30=5,

%e a(2)=6 because A051779(4)=32401 and sqrt(32401-1)/30=6.

%o (PARI) isok(n) = isprime(p = 30*n+1) && isprime(q = 30*n-1) && isprime(p*q+2); \\ _Michel Marcus_, Oct 11 2013

%Y Cf. A051779.

%K nonn

%O 1,1

%A _Zak Seidov_, Nov 26 2006