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

1,1

Comments

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

Links

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

Example

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

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 *)

Prog

(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", "))))}

Crossrefs

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

Cf. A001097.

Sequence in context: A056913 A002557 A128907 * A225709 A020162 A046404

Adjacent sequences: A321641 A321642 A321643 * A321645 A321646 A321647

Keyword

nonn,easy

Author

Dimitris Valianatos, Nov 15 2018

Status

approved