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

%I A361267 #34 Feb 16 2025 08:34:05
%S A361267 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 A361267 142,143,147,148,151,152,181,182,206,211,212,224,225,229,234,235,236,
%U A361267 253,261,264,276,285,286,287,301,302,313,314,322,332,336,352,384,389
%N A361267 Numbers k such that prime(k+2) - prime(k) = 6.
%H A361267 Robert Israel, <a href="/A361267/b361267.txt">Table of n, a(n) for n = 1..10000</a>
%H A361267 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeTriplet.html">Prime Triplet</a>
%H A361267 Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_triplet">Prime triplet</a>
%F A361267 a(n) = A000720(A007529(n)). - _Alois P. Heinz_, Mar 06 2023
%p A361267 q:= n-> is(ithprime(n+2)-ithprime(n)=6):
%p A361267 select(q, [$1..400])[];  # _Alois P. Heinz_, Mar 06 2023
%t A361267 Select[Range[400], Prime[# + 2] - Prime[#] == 6 &] (* _Michael De Vlieger_, Mar 06 2023 *)
%t A361267 PrimePi/@(Select[Partition[Prime[Range[400]],3,1],#[[3]]-#[[1]]==6&][[;;,1]]) (* _Harvey P. Dale_, Sep 16 2023 *)
%o A361267 (Clojure)
%o A361267 (defn next-prime [n]
%o A361267   (if (= n 2)
%o A361267       3
%o A361267       (let [m (+ n 2)
%o A361267             t (-> n Math/sqrt int (+ 2))]
%o A361267           (if (some #(zero? (mod m %)) (range 2 t))
%o A361267               (next-prime m)
%o A361267               m))))
%o A361267 (def primes (lazy-seq (iterate next-prime 2)))
%o A361267 (defn triplet-primes-positions [n]
%o A361267   (->> primes
%o A361267        (take n)
%o A361267        (partition 3 1)
%o A361267        (map list (range))
%o A361267        (filter (fn [[i xs]] (= 6 (- (last xs) (first xs)))))
%o A361267        (map #(-> % first inc))))
%o A361267 (println (triplet-primes-positions 2000))
%o A361267 (Python)
%o A361267 from itertools import count, islice
%o A361267 from sympy import nextprime, prime
%o A361267 def A361267_gen(startvalue=1): # generator of terms >= startvalue
%o A361267     p = prime(m:=max(startvalue,1))
%o A361267     q = nextprime(p)
%o A361267     r = nextprime(q)
%o A361267     for k in count(m):
%o A361267         if r-p == 6:
%o A361267             yield k
%o A361267         p, q, r = q, r, nextprime(r)
%o A361267 A361267_list = list(islice(A361267_gen(),20)) # _Chai Wah Wu_, Mar 27 2023
%Y A361267 Cf. A000040, A000720, A007529, A022004, A022005.
%K A361267 nonn
%O A361267 1,1
%A A361267 _Atabey Kaygun_, Mar 06 2023