A328822 Primes of the form p=3*q+3*r+q*r where q and r are distinct primes and 2*p-3*q, 2*p-3*r and 2*p-q*r are also prime.

103, 151, 439, 503, 727, 751, 887, 1063, 1091, 1223, 1447, 1951, 2791, 2903, 3079, 3191, 3371, 3491, 3851, 4567, 5051, 5431, 5591, 5647, 6151, 6491, 7591, 9463, 9623, 11171, 12911, 14891, 15511, 16183, 17203, 18787, 19031, 19403, 19991, 20231, 21863, 22111, 22391, 22567, 24251, 24407, 26951

Offset

1,1

Comments

The first term that occurs for more than one pair (q,r) is a(11)=1447, which corresponds to (q,r) = (5, 179) and (11, 101).

The first term that occurs for more than two pairs (q,r) is a(2579)=15108791, which corresponds to (q,r) = (17, 755437), (37, 377717), and (2797, 5393).

Links

Robert Israel, Table of n, a(n) for n = 1..6030

Example

a(3)=439 is in the sequence because q=5 and r=53 are distinct primes with 439=3*q+3*r+q*r and 439, 2*439-3*q=863, 2*439-3*r=719 and 2*439-q*r=613 are all primes.

Maple

N:= 10^5: # to get all terms <= N
Primes:= select(isprime, [seq(i, i=3..nextprime(N/8), 2)]):
filter:= proc(p, q, r)
isprime(p*q+2*p*r+2*q*r) and isprime(2*p*q+p*r+2*q*r) and isprime(2*p*q+2*p*r+q*r)
end proc:
p:= 3: R:= {}:
for iq from 2 do
   q:= Primes[iq];
   if 2*p*q + q^2 >= N then break fi;
   for ir from iq+1 do
       r:= Primes[ir];
       s:= p*q + q*r + p*r;
       if s > N then break fi;
       if isprime(s) and filter(p, q, r) then
         R:= R union {s};
       fi;
od od:
sort(convert(R, list));

Mathematica

M = 10^5; (* to get all terms <= M *)
filterQ[p_, q_, r_] := PrimeQ[p q + 2 p r + 2 q r] && PrimeQ[2 p q + p r + 2 q r] && PrimeQ[2 p q + 2 p r + q r];
primes = Select[Table[i, {i, 3, NextPrime[M/8], 2}], PrimeQ];
p = 3; R = {};
For[iq = 2, True, iq++, q = primes[[iq]]; If[2 p q + q^2 >= M, Break[]]; For[ir = iq + 1, True, ir++, r = primes[[ir]]; s = p q + q r + p r; If[s > M, Break[]]; If[PrimeQ[s] && filterQ[p, q, r], R = Union[R, {s}]]]];
R (* Jean-François Alcover, Aug 12 2020, after Robert Israel *)

Crossrefs

Primes in A328805.

Sequence in context: A127027 A033256 A106983 * A141962 A309412 A115267

Adjacent sequences: A328819 A328820 A328821 * A328823 A328824 A328825

Keyword

nonn

Author

Robert Israel, Oct 28 2019

Status

approved