A361267 Numbers k such that prime(k+2) - prime(k) = 6.

3, 4, 5, 6, 7, 12, 13, 19, 25, 26, 27, 28, 43, 44, 48, 49, 59, 63, 64, 69, 88, 89, 112, 116, 142, 143, 147, 148, 151, 152, 181, 182, 206, 211, 212, 224, 225, 229, 234, 235, 236, 253, 261, 264, 276, 285, 286, 287, 301, 302, 313, 314, 322, 332, 336, 352, 384, 389

Offset: 1

Author

Atabey Kaygun, Mar 06 2023

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Triplet
• Wikipedia, Prime triplet

Crossrefs

Cf. A000040, A000720, A007529, A022004, A022005.

Programs

Clojure

(defn next-prime [n]
  (if (= n 2)
      3
      (let [m (+ n 2)
            t (-> n Math/sqrt int (+ 2))]
          (if (some #(zero? (mod m %)) (range 2 t))
              (next-prime m)
              m))))
(def primes (lazy-seq (iterate next-prime 2)))
(defn triplet-primes-positions [n]
  (->> primes
       (take n)
       (partition 3 1)
       (map list (range))
       (filter (fn [[i xs]] (= 6 (- (last xs) (first xs)))))
       (map #(-> % first inc))))
(println (triplet-primes-positions 2000))

Maple

q:= n-> is(ithprime(n+2)-ithprime(n)=6):
select(q, [$1..400])[];  # Alois P. Heinz, Mar 06 2023

Mathematica

Select[Range[400], Prime[# + 2] - Prime[#] == 6 &] (* Michael De Vlieger, Mar 06 2023 *)
PrimePi/@(Select[Partition[Prime[Range[400]],3,1],#[[3]]-#[[1]]==6&][[;;,1]]) (* Harvey P. Dale, Sep 16 2023 *)

Python

from itertools import count, islice
from sympy import nextprime, prime
def A361267_gen(startvalue=1): # generator of terms >= startvalue
    p = prime(m:=max(startvalue,1))
    q = nextprime(p)
    r = nextprime(q)
    for k in count(m):
        if r-p == 6:
            yield k
        p, q, r = q, r, nextprime(r)
A361267_list = list(islice(A361267_gen(),20)) # Chai Wah Wu, Mar 27 2023

Formula

a(n) = A000720(A007529(n)). - Alois P. Heinz, Mar 06 2023