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A383243 Primes of the form p(k)*p(k+1)*(p(k+1) - p(k)) - 1 sorted by increasing k.

%I A383243 #16 May 15 2025 21:41:22
%S A383243 5,29,307,883,1747,4001,6067,26227,108883,152083,424481,311347,396883,
%T A383243 848201,580627,1713709,1814509,864883,5092973,3046789,3386989,1664083,
%U A383243 2581961,2196307,2304307,2377747,6955309,3526867,4088467,20916053,4796083,7339361
%N A383243 Primes of the form p(k)*p(k+1)*(p(k+1) - p(k)) - 1 sorted by increasing k.
%C A383243 Conjecture: there are infinitely many such primes.
%p A383243 q:= 2; R:= NULL: count:= 0:
%p A383243 while count < 100 do
%p A383243   p:= q;
%p A383243   q:= nextprime(q);
%p A383243   v:= p*q*(q-p)-1;
%p A383243   if isprime(v) then R:= R,v; count:= count+1 fi;
%p A383243 od:
%p A383243 R; # _Robert Israel_, May 11 2025
%t A383243 z = 200; p[n_] := Prime[n];
%t A383243 f[n_] := p[n]*p[n + 1]*(p[n + 1] - p[n])
%t A383243 t1 = Table[f[n] - 1, {n, 1, z}];    (* A383241 *)
%t A383243 t2 = Table[f[n] + 1, {n, 1, z}];    (* A383242 *)
%t A383243 Select[t1, PrimeQ[#] &]  (* A383243 *)
%t A383243 Select[t2, PrimeQ[#] &]  (* A383244 *)
%Y A383243 Cf. A000042, A383242, A383244.
%Y A383243 Primes in A383241.
%K A383243 nonn
%O A383243 1,1
%A A383243 _Clark Kimberling_, May 07 2025