A355803 Lower twin primes p such that p*(p+2)+p-1, p*(p+2)+p+1 and p*(p+2)+p+3 have at most four distinct prime factors between them.

3, 5, 137, 5639, 13397, 33599, 178247, 181607, 255467, 380867, 415379, 439007, 486677, 670037, 931727, 970787, 1163717, 1244987, 1259537, 1343057, 1384067, 1453547, 1540709, 1596347, 1619417, 1793357, 1908659, 2027897, 2097479, 2161637, 2276999, 2840777, 3163967, 3327167, 3536789, 3633347

Offset

1,1

Comments

One of the prime factors must be 3.

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

a(3) = 137 is a term because 137 and 139 are primes and 137*139+137-1 = 19179 = 3^2 * 2131, 137*139+137+1 = 19181 and 137*139+137+3 = 19183 have the four prime factors 3, 2131, 19181 and 19183.

Maple

R:= 3: count:= 1:
for p from 5 by 6 while count < 40 do
  if isprime(p) and isprime(p+2) and nops(numtheory:-factorset(p^2+3*p-1) union numtheory:-factorset(p^2+3*p+1)
    union numtheory:-factorset(p^2+3*p+3)) <= 4 then
      count:= count+1; R:= R, p;
  fi;
od:
R;

Mathematica

Select[Prime[Range[250000]], PrimeQ[# + 2] && Length[Union[(Join @@ FactorInteger[#^2 + 3*# + {-1, 1, 3}])[[;; , 1]]]] <= 4 &] (* Amiram Eldar, Jul 20 2022 *)

Crossrefs

Cf. A001359.

Sequence in context: A088269 A164371 A225672 * A240125 A108013 A087307

Adjacent sequences: A355800 A355801 A355802 * A355804 A355805 A355806

Keyword

nonn

Author

J. M. Bergot and Robert Israel, Jul 17 2022

Status

approved