A007529 -id:A007529 - cp's OEIS frontend

A357052 Distance from 10^n to the next prime triplet.

4, 1, 1, 87, 267, 357, 33, 451, 2011, 2821, 10687, 2497, 5073, 5557, 15243, 7147, 7357, 7197, 6627, 9157, 26317, 25833, 39207, 56067, 6667, 32937, 70561, 106533, 597, 28503, 19167, 74551, 301711, 6747, 246871, 223353, 63057, 75183, 48513, 61323, 16107, 554287, 160141, 29821, 220711, 49441

Offset: 0

M. F. Hasler, Sep 14 2022

Equivalently, least k > 0 such that either 10^n + k + {0, 2, 6} or 10^n + k + {0, 4, 6} are primes.

The initial term, index n = 0, is the only even term and the only case where the last member of the triplet has one digit more than the first member. The value a(0) = 4 correspond to the prime triplet (5, 7, 11). We do not consider the triplets (2, 3, 5) or (3, 5, 7) which come earlier but do not follow the standard pattern.

			(11, 13, 17) and (101, 103, 107) are the smallest 2-digit and 3-digit prime triplets, at distance a(1) = a(2) = 1 from 10^1 and 10^2, respectively.
(1087, 1091, 1093) is the smallest 4-digit prime triplet, at distance a(3) = 87 from 10^3.
a(6999) = 1141791245437 is the distance from 10^6999 to the smallest 7000 digit prime triplet, of the form (p, p+2, p+6).

Michael S. Branicky, Table of n, a(n) for n = 0..216
Norman Luhn, smallest 7000 digit prime triplet, primenumberstheory mailing list at groups.io, Sep 14 2022

Cf. A007529 (start of prime triplets), A022004 and A022005 (start of prime triples {0,2,6} resp. {0,4,6}), A343635 (same for quintuplets).

f:= proc(n) local p;
   for p from 10^n + 1 by 2 do
     if p mod 3 = 1 then if isprime(p) and isprime(p+4) and isprime(p+6) then return p-10^n fi
     elif p mod 3 = 2 and isprime(p) and isprime(p+2) and isprime(p+6) then
return p-10^n
     fi
   od;
end proc:
f(0):= 4:
map(f, [$0..45]); # Robert Israel, Sep 15 2022
A357052 := proc(n) local p,q,r; p,q,r := 10^n,0,0; while p-r <> 6 do p,q,r := nextprime(p+1),p,q; od; r-10^n; end; # M. F. Hasler, Sep 15 2022

A357052(n,q=-9,r=-9)=forprime(p=10^n,,p-r<7 && return(r-10^n);[q,r]=[p,q])

a(n) = min{ k>0 | 10^n + k + [0, 6] contains 3 primes }.

a(n) = min A007529 ∩ [10^n, oo) for n > 0.

A055737 Number of prime triples < 10^n, where a prime triple means 3 successive primes of the form {p, p+2, p+4} or {p, p+4, p+6}.

Original entry on oeis.org

0, 8, 30, 112, 507, 2837, 17220, 111156, 759256, 5425573, 40174725, 305689269, 2379622234, 18887841658

Offset: 1

Robert G. Wilson v, Jun 09 2000

For this sequence all three members of the triple must be below the 10^n bound. - Sean A. Irvine, Apr 04 2022

J. Recreational Math., vol. 23, No. 2, 1991, p. 97.

Index entries for sequences related to numbers of primes in various ranges

Cf. A055738, A006880, A007529, A098415.

x=168; a=Table[ Prime[ n ], {n, 1, x} ] c=0; Do[ If[ a[[ n ]]+6==a[ [ n+2 ] ], c++ ], {n, 1, x-2} ]; c # the values of x to use are given by A006880

a(7)-a(9) from Jud McCranie, Oct 07 2000
a(10)-a(12) from Martin Raab, Oct 04 2006
a(13)-a(14) from Charles R Greathouse IV, Feb 09 2022

A143726 Middle members of beastly cousin prime triples: primes p such that both p+666 and p-666 are prime.

Original entry on oeis.org

733, 773, 823, 857, 877, 947, 997, 1033, 1087, 1123, 1213, 1223, 1283, 1307, 1327, 1423, 1487, 1607, 1993, 2027, 2137, 2153, 2237, 2273, 2287, 2333, 2543, 2663, 2677, 2693, 2797, 2803, 2917, 3187, 3257, 3323, 3407, 3433, 3463, 3467, 3593, 3623, 3847

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 30 2008

nonn

			733 - 666 = 67, 733 + 666 = 1399 and 67, 733, 1399 are all prime, so 733 is a term of the sequence. - _Felix Fröhlich_, May 26 2022

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A007529 (p, p+2 or +4, p+6 prime), A023200 (p and p+4 prime), A046132 (p-4 and p prime), A073648 (p-2, p and p+4 prime).

lst={};b=666;Do[p=Prime[n];If[PrimeQ[p+b]&&PrimeQ[p-b],AppendTo[lst,p]],{n,5!+2,7!}];lst
Select[Prime[Range[122,600]],AllTrue[#+{666,-666},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 08 2018 *)

forprime(p=1, 3900, if(ispseudoprime(p+666) && ispseudoprime(p-666), print1(p, ", "))) \\ Felix Fröhlich, May 26 2022

Name edited by Felix Fröhlich, May 26 2022

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

Original entry on oeis.org

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

Atabey Kaygun, Mar 06 2023

nonn

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

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

(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))

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

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 *)

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

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

A275683 Difference between the smallest 10^n-digit member of a prime triple and 10^(10^n - 1).

Original entry on oeis.org

4, 2821, 1821127, 1598241813

Offset: 0

Arkadiusz Wesolowski, Aug 05 2016

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios!: 10000...21127 (100-digits)
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios!: 10000...41813 (1000-digits)

Cf. A007529.

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A357052 Distance from 10^n to the next prime triplet.

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A055737 Number of prime triples < 10^n, where a prime triple means 3 successive primes of the form {p, p+2, p+4} or {p, p+4, p+6}.

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A143726 Middle members of beastly cousin prime triples: primes p such that both p+666 and p-666 are prime.

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

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A275683 Difference between the smallest 10^n-digit member of a prime triple and 10^(10^n - 1).

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