A347531 - OEIS

%I #48 Mar 14 2022 08:46:22

%S 5,19,31,97,107,1429,1597,1997,2441,2917,3203,3581,3659,3691,3847,

%T 7481,8237,8377,8737,9697,11467,12107,12251,14071,17317,17659,18481,

%U 18583,21383,21491,23189,24799,30931,33563,34337,37507,38317,38639,38707,39397,40483,41221,43577,43649,45121,46411,51059

%N First of three consecutive primes p,q,r such that r^2-p^2+p, r^2-p^2+q and r^2-p^2+r are consecutive primes.

%C Dickson's conjecture implies that there are infinitely many terms with, for example, the first three consecutive primes p, p+2, p+6 and the second three 13*p+36, 13*p+38 and 13*p+42.

%H Robert Israel, <a href="/A347531/b347531.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3) = 31 is a term because 31, 37, 41 are consecutive primes and 41^2-31^2+31 = 751, 41^2-31^2+37 = 757, 41^2-31^2+41 = 761 are consecutive primes.

%p R:= NULL: count:= 0:

%p q:= 2: r:= 3:

%p while count < 50 do

%p   p:= q; q:= r; r:= nextprime(r);

%p   d:= r^2 - p^2;

%p   if isprime(p+d) and nextprime(p+d)=q+d and nextprime(q+d)=r+d then

%p     count:= count+1; R:= R, p

%p   fi

%p od:

%p R;

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Mar 10 2022