A321644 Squarefree odd composite numbers whose factors are all twin primes (not necessarily from the same pair).

15, 21, 33, 35, 39, 51, 55, 57, 65, 77, 85, 87, 91, 93, 95, 105, 119, 123, 129, 133, 143, 145, 155, 165, 177, 183, 187, 195, 203, 205, 209, 213, 215, 217, 219, 221, 231, 247, 255, 273, 285, 287, 295, 301, 303, 305, 309, 319, 321, 323, 327, 341, 355, 357, 365

Offset: 1

Author

Dimitris Valianatos, Nov 15 2018

Keywords

• nonn
• easy

Comments

This sequence has infinitely many terms if and only if the twin prime conjecture is true.

Examples

			a(3) = 33 = 3 * 11; 3 and 11 are both twin primes, but not from the same pair.
		

Links

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

Crossrefs

Subsequence of A024556, and hence of A056911, A061346, and A071904.

Cf. A001097.

Programs

Maple

N:= 1000: # to get all terms <= N
P:= select(isprime, {seq(i,i=3..(N+6)/3,2)}):
TP:= P intersect map(`-`,P,2):
TP:= TP union map(`+`,TP,2):
Agenda:= map(t -> [t],TP): Res:= NULL:
while Agenda <> {} do
   Agenda:= map(proc(t) local s; seq([op(t),s], s = select(s -> s > t[-1] and s*convert(t,`*`) <= N , TP)) end proc, Agenda);
   Res:= Res, op(map(convert,Agenda,`*`));
od:
sort([Res]); # Robert Israel, Jan 27 2019

Mathematica

seqQ[n_] := CompositeQ[n] && SquareFreeQ[n] && Module[{f = FactorInteger[n][[;;, 1]]}, Length[Select[f, PrimeQ[# - 2] || PrimeQ[# + 2] &]] == Length[f]]; Select[ Range[1, 365, 2], seqQ] (* Amiram Eldar, Nov 15 2018 *)

PARI

{forcomposite(n=3, 1000, if(moebius(n) <> 0, v = factor(n)~; i = 0;for(k = 1, #v,p=v[1,k]; if(isprime(p-2)||isprime(p+2), i++));if(i==#v,print1(n", "))))}