A358132 Numbers k such that there exists a pair of primes (p,q) with p+q = k such that p*q + k, p*q - k, p*q + A001414(k) and p*q - A001414(k) are all prime.

7, 60, 72, 114, 186, 378, 474, 480, 762, 884, 1266, 1338, 1374, 2004, 2742, 3012, 3234, 3246, 3276, 3282, 3618, 3936, 4230, 4620, 5154, 5514, 5544, 5724, 5886, 6006, 6054, 6084, 6234, 6306, 6414, 6510, 6522, 6762, 6774, 6858, 7410, 7422, 7764, 8286, 8394, 8538, 8688, 8826, 8892, 8916, 9186, 9264

Offset

1,1

Comments

Most terms == 0 (mod 6), but there are terms such as 7 and 15585 where p = 2, and

884 and 10010 where p = 3.

Conjecture: the sequence includes all sufficiently large k such that both k and A001414(k) are divisible by 6.

Links

Robert Israel, Table of n, a(n) for n = 1..828

Example

a(5) = 186 is a term because p = 89 and q = 97 are primes with 89 + 97 = 186, 186 = 2*3*31 so A001414(186) = 2+3+31 = 36, and 89*97 + 186 = 8819, 89*97 - 186 = 8447, 89*97 + 36 =  8669 and 89*97 - 36 = 8597 are all primes.

Maple

spf:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
filter:= proc(k) local p, q, pq, s;
  s:= spf(k);
  if k::odd then
    q:= k-2;
    return (s::odd and isprime(q) and isprime(2*q+s) and isprime(2*q-s) and isprime(2*q+k) and isprime(2*q-k))
  fi;
if k mod 6 <> 0 then
    q:= k-3;
    return (isprime(q) and isprime(3*q+s) and isprime(3*q-s) and isprime(3*q+k) and isprime(3*q-k))
  fi;
  p:= 2;
  do
    p:= nextprime(p);
    q:= k-p;
    if p > q then return false fi;
    pq:= p*q;
    if isprime(q) and isprime(pq+k) and isprime(pq-k) and isprime(pq+s) and isprime(pq-s) then return true fi
  od
end proc:
select(filter, [$1..10000]);

Crossrefs

Cf. A001414.

Sequence in context: A203174 A183260 A285227 * A063969 A366613 A303120

Adjacent sequences: A358129 A358130 A358131 * A358133 A358134 A358135

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Oct 31 2022

Status

approved