A328805 Numbers of the form k = p*q + p*r + q*r where p < q < r are distinct primes such that 2*k-p*q, 2*k-p*r and 2*k-q*r are prime.

103, 119, 151, 327, 355, 439, 451, 503, 511, 583, 711, 723, 727, 751, 791, 887, 1063, 1091, 1119, 1175, 1223, 1251, 1447, 1551, 1647, 1751, 1951, 1991, 2071, 2151, 2583, 2651, 2743, 2775, 2791, 2903, 2915, 2951, 2967, 3075, 3079, 3171, 3191, 3311, 3335, 3367, 3371, 3435, 3491, 3575, 3579, 3651

Offset

1,1

Comments

The first term that occurs for two different triples (p,q,r) is 791, which corresponds to (p,q,r) = (3,5,97) and (3,17,37).

The first term that occurs for three different triples (p,q,r) is 66135, which corresponds to (p,q,r) = (11,71,797), (17,29,1427) and (17,59,857).

All terms == 3 (mod 4).

If p <> 3, then p,q,r are all congruent mod 6 so k is divisible by 3.

If 5 is not p or q, then two of (p,q,r) are congruent to each other mod 10.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(3) = 151 is in the sequence because (p,q,r)=(3,7,13) are distinct primes such that p*q+p*r+q*r=151 and 2*151-p*q=281, 2*151-p*r=263 and 2*151-q*r=211 are primes.

Maple

N:= 4000: # to get all terms <= N
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:
Primes:= select(isprime, [seq(i, i=3..N/8, 2)]):
R:= {}:
for ip from 1 do
  p:= Primes[ip];
  if 3*p^2 >= N then break fi;
  for iq from ip+1 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 filter(p, q, r) then
         R:= R union {s};
       fi;
od od od:
sort(convert(R, list));

Mathematica

M = 4000; (* 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, M/8, 2}], PrimeQ];
R = {};
For[ip = 1, True, ip++, p = primes[[ip]]; If[3 p^2 >= M, Break[]]; For[iq = ip + 1, 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[filterQ[p, q, r], R = Union[R, {s}]]]]];
R (* Jean-François Alcover, Jul 31 2020, after Robert Israel *)

Crossrefs

Cf. A328822 (primes in this sequence).

Sequence in context: A140817 A274518 A066131 * A095639 A193143 A098049

Adjacent sequences: A328802 A328803 A328804 * A328806 A328807 A328808

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Oct 27 2019

Status

approved