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A383244 Primes of the form p(k)*p(k+1)*(p(k+1) - p(k)) + 1 sorted by increasing k.

%I A383244 #16 May 15 2025 21:41:03
%S A383244 7,31,71,647,4003,6883,3527,14947,34603,20807,23327,173347,73727,
%T A383244 503869,103967,145799,450403,194687,669283,848203,1193443,1775563,
%U A383244 649799,1976803,2088547,2131243,4687069,2534947,2581963,5338237,3250123,3411043,1555847,5346763
%N A383244 Primes of the form p(k)*p(k+1)*(p(k+1) - p(k)) + 1 sorted by increasing k.
%C A383244 Conjecture: there are infinitely many such primes.
%p A383244 =q:= 2; R:= NULL: count:= 0:
%p A383244 while count < 100 do
%p A383244   p:= q;
%p A383244   q:= nextprime(q);
%p A383244   v:= p*q*(q-p)+1;
%p A383244   if isprime(v) then R:= R,v; count:= count+1 fi;
%p A383244 od:
%p A383244 R; # _Robert Israel_, May 11 2025
%t A383244 z = 200; p[n_] := Prime[n];
%t A383244 f[n_] := p[n]*p[n + 1]*(p[n + 1] - p[n])
%t A383244 t1 = Table[f[n] - 1, {n, 1, z}];    (* A383241 *)
%t A383244 t2 = Table[f[n] + 1, {n, 1, z}];    (* A383242 *)
%t A383244 Select[t1, PrimeQ[#] &]  (* A383243 *)
%t A383244 Select[t2, PrimeQ[#] &]  (* A383244 *)
%o A383244 (PARI) select(isprime, vector(200, k, prime(k)*prime(k+1)*(prime(k+1) - prime(k)) + 1)) \\ _Michel Marcus_, May 12 2025
%Y A383244 Cf. A000042, A383241, A383243.
%Y A383244 Primes in A383242.
%K A383244 nonn
%O A383244 1,1
%A A383244 _Clark Kimberling_, May 07 2025