%I #34 Feb 16 2025 08:34:05

%S 3,4,5,6,7,12,13,19,25,26,27,28,43,44,48,49,59,63,64,69,88,89,112,116,

%T 142,143,147,148,151,152,181,182,206,211,212,224,225,229,234,235,236,

%U 253,261,264,276,285,286,287,301,302,313,314,322,332,336,352,384,389

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

%H Robert Israel, <a href="/A361267/b361267.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeTriplet.html">Prime Triplet</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_triplet">Prime triplet</a>

%F a(n) = A000720(A007529(n)). - _Alois P. Heinz_, Mar 06 2023

%p q:= n-> is(ithprime(n+2)-ithprime(n)=6):

%p select(q, [$1..400])[]; # _Alois P. Heinz_, Mar 06 2023

%t Select[Range[400], Prime[# + 2] - Prime[#] == 6 &] (* _Michael De Vlieger_, Mar 06 2023 *)

%t PrimePi/@(Select[Partition[Prime[Range[400]],3,1],#[[3]]-#[[1]]==6&][[;;,1]]) (* _Harvey P. Dale_, Sep 16 2023 *)

%o (Clojure)

%o (defn next-prime [n]

%o (if (= n 2)

%o 3

%o (let [m (+ n 2)

%o t (-> n Math/sqrt int (+ 2))]

%o (if (some #(zero? (mod m %)) (range 2 t))

%o (next-prime m)

%o m))))

%o (def primes (lazy-seq (iterate next-prime 2)))

%o (defn triplet-primes-positions [n]

%o (->> primes

%o (take n)

%o (partition 3 1)

%o (map list (range))

%o (filter (fn [[i xs]] (= 6 (- (last xs) (first xs)))))

%o (map #(-> % first inc))))

%o (println (triplet-primes-positions 2000))

%o (Python)

%o from itertools import count, islice

%o from sympy import nextprime, prime

%o def A361267_gen(startvalue=1): # generator of terms >= startvalue

%o p = prime(m:=max(startvalue,1))

%o q = nextprime(p)

%o r = nextprime(q)

%o for k in count(m):

%o if r-p == 6:

%o yield k

%o p, q, r = q, r, nextprime(r)

%o A361267_list = list(islice(A361267_gen(),20)) # _Chai Wah Wu_, Mar 27 2023

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

%K nonn

%O 1,1

%A _Atabey Kaygun_, Mar 06 2023