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Primes of the form pq + p + 1 where p < q are adjacent primes.
2

%I #15 Sep 08 2022 08:45:54

%S 19,41,457,691,929,2539,3181,3659,6637,19181,29059,32579,55921,57839,

%T 60733,71011,83203,123547,127081,132113,149371,154823,176819,181873,

%U 194917,245501,320911,382541,430981,497711,510481,523433,549817,595207

%N Primes of the form pq + p + 1 where p < q are adjacent primes.

%C In the sequence of first differences D(9) = 12544 is the square of 112.

%C In the sequence of second differences DD(3) = 4 and DD(7) = 2500 are squares.

%H Harvey P. Dale, <a href="/A180932/b180932.txt">Table of n, a(n) for n = 1..10000</a>

%e a(7) = 3181 since 3181 = 53*(59+1) + 1 where 53 and 59 are successive primes.

%t Select[Times@@#+#[[1]]+1&/@Partition[Prime[Range[200]],2,1],PrimeQ] (* _Harvey P. Dale_, Apr 26 2014 *)

%o (PARI) p=2;forprime(q=3,1e3,k=p*(q+1)+1;if(isprime(k),print1(k","));p=q) \\ _Charles R Greathouse IV_, Sep 27 2010

%o (Magma) [a: n in [0..200] | IsPrime(a) where a is NthPrime(n)*NthPrime(n+1)+NthPrime(n)+1]; // _Vincenzo Librandi_, Jul 07 2017

%Y Cf. A180931.

%Y Primes in A286624.

%K nonn

%O 1,1

%A _Carmine Suriano_, Sep 26 2010

%E New description and editing from _Charles R Greathouse IV_, Sep 27 2010