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

7, 31, 71, 647, 4003, 6883, 3527, 14947, 34603, 20807, 23327, 173347, 73727, 503869, 103967, 145799, 450403, 194687, 669283, 848203, 1193443, 1775563, 649799, 1976803, 2088547, 2131243, 4687069, 2534947, 2581963, 5338237, 3250123, 3411043, 1555847, 5346763

Offset: 1

Author

Clark Kimberling, May 07 2025

Keywords

• nonn

Comments

Conjecture: there are infinitely many such primes.

Crossrefs

Cf. A000042, A383241, A383243.

Primes in A383242.

Programs

Maple

=q:= 2; R:= NULL: count:= 0:
while count < 100 do
  p:= q;
  q:= nextprime(q);
  v:= p*q*(q-p)+1;
  if isprime(v) then R:= R,v; count:= count+1 fi;
od:
R; # Robert Israel, May 11 2025

Mathematica

z = 200; p[n_] := Prime[n];
f[n_] := p[n]*p[n + 1]*(p[n + 1] - p[n])
t1 = Table[f[n] - 1, {n, 1, z}];    (* A383241 *)
t2 = Table[f[n] + 1, {n, 1, z}];    (* A383242 *)
Select[t1, PrimeQ[#] &]  (* A383243 *)
Select[t2, PrimeQ[#] &]  (* A383244 *)

PARI

select(isprime, vector(200, k, prime(k)*prime(k+1)*(prime(k+1) - prime(k)) + 1)) \\ Michel Marcus, May 12 2025