A007811 -id:A007811 - cp's OEIS frontend

A007530 Prime quadruples: numbers k such that k, k+2, k+6, k+8 are all prime.

5, 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, 5651, 9431, 13001, 15641, 15731, 16061, 18041, 18911, 19421, 21011, 22271, 25301, 31721, 34841, 43781, 51341, 55331, 62981, 67211, 69491, 72221, 77261, 79691, 81041, 82721, 88811, 97841, 99131

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

Except for the first term, 5, all terms == 11 (mod 30). - Zak Seidov, Dec 04 2008
Some further values: For k = 1, ..., 10, a(k*10^3) = 11721791, 31210841, 54112601, 78984791, 106583831, 136466501, 165939791, 196512551, 230794301, 265201421. - M. F. Hasler, May 04 2009
k is the first prime of 2 consecutive twin prime pairs. - Daniel Forgues, Aug 01 2009
The prime quadruples of form p + (0, 2, 6, 8) have the quadruple congruence class (-1, +1, -1, +1) (mod 6). - Daniel Forgues, Aug 12 2009
s = (p+8)-(p) = 8 is the smallest s giving an admissible prime quadruple form, for which the only admissible form is p + (0, 2, 6, 8), since (0, 2, 6, 8) is the only form not covering all the congruence classes for any prime <= 4. Since s is smallest, these prime quadruples are prime constellations (or prime quadruplets), i.e., they contain consecutive primes. - Daniel Forgues, Aug 12 2009
Except for the first term, 5, all prime quadruples are of the form (15k-4, 15k-2, 15k+2, 15k+4), with k >= 1, and so are centered on 15k. - Daniel Forgues, Aug 12 2009
Subsequence of A022004. - R. J. Mathar, Feb 10 2013
The quadruplets are listed in A136162. - M. F. Hasler, Apr 20 2013
Starting at a(2) and examining the first 50 terms, (a(n)+4)/15 is a prime in 8 cases and a semiprime in 21; the last 18 terms have 2 primes and 11 semiprimes. Do the number of semiprimes continue to occur greater than mere chance? - J. M. Bergot, Apr 27 2015

			From _M. F. Hasler_, May 04 2009: (Start)
a(1)=5 is the start of the first prime quadruplet, {5,7,11,13}.
a(2)=11 is the start of the second prime quadruplet, {11,13,17,19}, and all other prime quadruplets differ from this one by a multiple of 30.
a(100)=470081 is the start of the 100th prime quadruplet;
a(500)=4370081 is the start of the 500th prime quadruplet.
a(167)=1002341 is the least quadruplet prime beyond 10^6. (End)

H. Rademacher, Lectures on Elementary Number Theory. Blaisdell, NY, 1964, p. 4.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Matt C. Anderson, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe).
C. K. Caldwell, The Prime Glossary, prime quadruple
Tony Forbes and Norman Luhn, prime k-tuplets
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Norman Luhn, Table of n, a(n) for n = 1..1000000
Thomas R. Nicely, Enumeration to 1.6e15 of the prime quadruplets
H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985, ISBN: 978-0-8176-8297-2, Chap. 4, see p. 65.
Eric Weisstein's World of Mathematics, Prime Quadruplet

Cf. A159910 (first differences divided by 30), A120120, A007811, A014561.

[ p: p in PrimesUpTo(11000)| IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8)] // Vincenzo Librandi, Nov 18 2010

A007530 = Select[Range[1, 10^5 - 1, 2], Union[PrimeQ[# + {0, 2, 6, 8}]] == {True} &] (* Alonso del Arte, Sep 24 2011 *)
Select[Prime[Range[10000]],AllTrue[#+{2,6,8},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 11 2019 *)

A007530( n, print_all=0, s=2 )={ my(p,q,r); until(!n--, until( p+8==s=nextprime(s+2), p=q; q=r; r=s); print_all && print1(p","));p} \\ The optional 3rd argument can be used to obtain large values by starting from some precomputed point instead of zero, using a(n+k) = A007530(k+1,,a(n)) (or A007530(k,,a(n)-1) for k>0); e.g., you get a(10^4+k) using A007530(k+1,,265201421) (value of a(10^4) from the comments section). - M. F. Hasler, May 04 2009

forprime(p=2, 10^5, if(isprime(p+2) && isprime(p+6) && isprime(p+8), print1(p, ", "))) \\ Felix Fröhlich, Jun 22 2014

from sympy import primerange
def aupto(limit):
  p, q, r, alst = 2, 3, 5, []
  for s in primerange(7, limit+9):
    if p+2 == q and p+6 == r and p+8 == s: alst.append(p)
    p, q, r = q, r, s
  return alst
print(aupto(10**5)) # Michael S. Branicky, May 11 2021

a(n) = 11 + 30*A014561(n-1) for n > 1. - M. F. Hasler, May 04 2009

More terms from Warut Roonguthai
Incorrect formula and Mathematica program removed by N. J. A. Sloane, Dec 04 2008, at the suggestion of Zak Seidov
Values up to a(1000) checked with the given PARI code by M. F. Hasler, May 04 2009

A032352 Numbers k such that there is no prime between 10k and 10k+9.

Original entry on oeis.org

20, 32, 51, 53, 62, 84, 89, 107, 113, 114, 126, 133, 134, 135, 141, 146, 150, 164, 167, 168, 171, 176, 179, 185, 189, 192, 196, 204, 207, 210, 218, 219, 232, 236, 240, 248, 249, 251, 256, 258, 282, 294, 298, 305, 309, 314, 315, 317, 323, 324, 326, 328, 342

Offset: 1

Jeff Burch

Numbers k with property that appending any single decimal digit to k does not produce a prime.

A007920(n*10) > 10.

			m=32: 321=3*107, 323=17*19, 325=5*5*13, 327=3*109, 329=7*47, therefore 32 is a term.

M. I. Wilczynski, Table of n, a(n) for n = 1..10000

Cf. A124665 (subsequence), A010051, A007811, A216292, A216293.

a032352 n = a032352_list !! (n-1)
a032352_list = filter
   (\x -> all (== 0) $ map (a010051 . (10*x +)) [1..9]) [1..]
-- Reinhard Zumkeller, Oct 22 2011

[n: n in [1..350] | IsZero(#PrimesInInterval(10*n, 10*n+9))]; // Bruno Berselli, Sep 04 2012

a:=proc(n) if map(isprime,{seq(10*n+j,j=1..9)})={false} then n else fi end: seq(a(n),n=1..350); # Emeric Deutsch, Aug 01 2005

f[n_] := PrimePi[10n + 10] - PrimePi[10n]; Select[ Range[342], f[ # ] == 0 &] (* Robert G. Wilson v, Sep 24 2004 *)
Select[Range[342], NextPrime[10 # ] > 10 # + 9 &] (* Maciej Ireneusz Wilczynski, Jul 18 2010 *)
Flatten@Position[10*#+{1,3,7,9}&/@Range@4000,{?CompositeQ ..}] (* _Hans Rudolf Widmer, Jul 06 2024 *)

is(n)=!isprime(10*n+1) && !isprime(10*n+3) && !isprime(10*n+7) && !isprime(10*n+9) \\ Charles R Greathouse IV, Mar 29 2013

from sympy import isprime
def aupto(limit):
  alst = []
  for d in range(2, limit+1):
    td = [10*d + j for j in [1, 3, 7, 9]]
    if not any(isprime(t) for t in td): alst.append(d)
  return alst
print(aupto(342)) # Michael S. Branicky, May 30 2021

a(n) ~ n. - Charles R Greathouse IV, Mar 29 2013

More terms from Miklos Kristof, Aug 27 2002

A014561 Numbers k giving rise to prime quadruples (30k+11, 30k+13, 30k+17, 30k+19).

Original entry on oeis.org

0, 3, 6, 27, 49, 62, 69, 108, 115, 188, 314, 433, 521, 524, 535, 601, 630, 647, 700, 742, 843, 1057, 1161, 1459, 1711, 1844, 2099, 2240, 2316, 2407, 2575, 2656, 2701, 2757, 2960, 3261, 3304, 3370, 3661, 3884, 3976, 4073, 4515, 4805, 5242, 5523, 5561, 5705

Offset: 1

Eric W. Weisstein

Intersection of A089160 and A089161. - Zak Seidov, Dec 22 2006
This can be seen as a condensed version of A007530, which lists the first member of the actual prime quadruplet (30x+11, 30x+13, 30x+17, 30x+19), x=a(n). - M. F. Hasler, Dec 05 2013
Comment from Frank Ellermann, Mar 13 2020: (Start)
Ignoring 2 and 3, {5,7,11,13} is the only twin-twin prime quadruple not following this pattern for primes > 5. One candidate mod 30 corresponds to 7 candidates mod 210, but 7 * 7 = 30 + 19, 7 * 11 = 60 + 17, 7 * 19 = 120 + 13, and 7 * 23 = 190 + 11 are multiples of 7, leaving only 3 candidates mod 210.
Likewise, 13 * 13 = 150 + 19 is a multiple of 13 mod 30030, but 5 + 1001 * k is a proper subset of 5 + 7 * k with 1001 = 13 * 11 * 7. Other disqualified candidates with nonzero k are:
  13 * 17 = 210 + 11 for a(k) <> 7 + 1001 * k,
  11 * 29 = 300 + 19 for a(k) <> 10 + 77 * k,
  11 * 37 = 390 + 17 for a(k) <> 13 + 77 * k,
  19 * 23 = 420 + 17 for a(k) <> 14 + 321321 * k,
  17 * 31 = 510 + 17 for a(k) <> 17 + 17017 * k,
  13 * 47 = 600 + 11 for a(k) <> 20 + 1001 * k,
  11 * 59 = 630 + 19 for a(k) <> 21 + 77 * k, and
  11 * 67 = 720 + 17 for a(k) <> 24 + 77 + k, picking the smallest prime factors 11, 17, 11 for {407, 527, 737} instead of 13, 23, 17 for {403, 529, 731}.
(End)

			a(4) = 27 for 27*30 = 810 yields twin primes at 810+11 = A001359(32) = A000040(142) and 810+17 = A001359(33) = A000040(144) ending at 810+19 = A000040(145).

Michael De Vlieger, Table of n, a(n) for n = 1..10972 (first 1000 terms from Zak Seidov)
Eric Weisstein's World of Mathematics, Prime Quadruplet.

Cf. A089160, A089161.
Cf. A007530, A007811, A061668.
A100418 and A100423 are subsequences.

a014561Q[n_Integer] :=
  If[And[PrimeQ[30 n + 11], PrimeQ[30 n + 13], PrimeQ[30 n + 17],
     PrimeQ[30 n + 19]] == True, True, False];
a014561[n_Integer] :=
  Flatten[Position[Thread[a014561Q[Range[n]]], True]];
a014561[1000] (* Michael De Vlieger, Jul 17 2014 *)
Select[Range[0,6000],AllTrue[30#+{11,13,17,19},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 21 2016 *)

isok(n) = isprime(30*n+11) && isprime(30*n+13) && isprime(30*n+17) && isprime(30*n+19) \\ Michel Marcus, Jun 09 2013

a(n) = (A007811(n) - 1)/3. - Zak Seidov, Sep 21 2009
a(n) = (A007530(n+1) - 11)/30 = floor(A007530(n+1)/30). - M. F. Hasler, Dec 05 2013
a(n) = A061668(n) - 1. - Hugo Pfoertner, Nov 03 2023

More terms from Warut Roonguthai

A038800 Number of primes between 10n and 10n+9.

Original entry on oeis.org

4, 4, 2, 2, 3, 2, 2, 3, 2, 1, 4, 1, 1, 3, 1, 2, 2, 2, 1, 4, 0, 1, 3, 2, 1, 2, 2, 2, 2, 1, 1, 3, 0, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 2, 2, 0, 2, 0, 2, 1, 2, 2, 1, 2, 2, 3, 0, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 4, 1, 0, 3, 1, 1, 3, 0, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1

Offset: 0

Jeff Burch

If n runs through the primes, the subsequence 2, 2, 2, 3, 1, 3, 2, 4, 2, 1, 3, 2, 1, 3, 1, 0, 2, 3, 2,... is created. - R. J. Mathar, Jul 19 2012

Since 431, 433, and 439 are all prime, a(43)=3. - Bobby Jacobs, Sep 25 2016

Michael De Vlieger, Table of n, a(n) for n = 0..10000
A. Frank and P. Jacqueroux, International Contest, 2001. Numerators of Item 23

Cf. A055781, A055782, A055783, A055784.
Positions of 4's: {0} U A007811.
Cf. A098592.

Table[Count[Range[10 n, 10 n + 9], p_ /; PrimeQ@ p], {n, 0, 105}] (* Michael De Vlieger, Sep 25 2016 *)
Table[PrimePi[10n+9]-PrimePi[10n],{n,0,120}] (* Harvey P. Dale, May 04 2025 *)

a(n) = primepi(10*n+9) - primepi(10*n); \\ Michel Marcus, Sep 26 2016

a(43) corrected by Bobby Jacobs, Sep 25 2016

a(101) and a(104) corrected by Michael De Vlieger, Sep 25 2016

A067267 Numbers n such that n, 10n+1, 10n+3, 10n+7 and 10n+9 are all primes.

Original entry on oeis.org

19, 6949, 10111, 15727, 20149, 24799, 26041, 26881, 29587, 29947, 40213, 41203, 44257, 46747, 47701, 49057, 50023, 51061, 55921, 66109, 66643, 73939, 76819, 79579, 95947, 106501, 110083, 111781, 112213, 114679

Offset: 1

Sudipta Das (juitech(AT)vsnl.net), Feb 21 2002

nonn

All terms == 1 (mod 6). - Robert Israel, Jan 22 2018

			a(1)=19 because 191, 193, 197 and 199 are all prime.

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

Cf. A007811.

select(t -> andmap(isprime, [t,10*t+1,10*t+3,10*t+7,10*t+9]), [seq(i,i=1..2*10^5,6)]); # Robert Israel, Jan 22 2018

okQ[n_]:=Module[{n10=10n},And@@PrimeQ[{n10+1,n10+3,n10+7,n10+9}]]
Select[Prime[Range[20000]],okQ] (* Harvey P. Dale, Dec 13 2010 *)
Select[Prime[Range[11000]],AllTrue[10#+{1,3,7,9},PrimeQ]&] (* Harvey P. Dale, Feb 23 2022 *)

A173037 Numbers k such that k-4, k-2, k+2 and k+4 are prime.

Original entry on oeis.org

9, 15, 105, 195, 825, 1485, 1875, 2085, 3255, 3465, 5655, 9435, 13005, 15645, 15735, 16065, 18045, 18915, 19425, 21015, 22275, 25305, 31725, 34845, 43785, 51345, 55335, 62985, 67215, 69495, 72225, 77265, 79695, 81045, 82725, 88815, 97845

Offset: 1

Juri-Stepan Gerasimov, Feb 07 2010

nonn

Average k of the four primes in two twin prime pairs (k-4, k-2) and (k+2, k+4) which are linked by the cousin prime pair (k-2, k+2).
All terms are odd composites; except for a(1) they are multiples of 5.
Subsequence of A087679, of A087680 and of A164385.
All terms except for a(1) are multiples of 15. - Zak Seidov, May 18 2014
One of (k-1, k, k+1) is always divisible by 7. - Fred Daniel Kline, Sep 24 2015
Terms other than a(1) must be equivalent to 1 mod 2, 0 mod 3, 0 mod 5, and 0,+/-1 mod 7. Taken together, this requires terms other than a(1) to have the form 210k+/-15 or 210k+105. However, not all numbers of that form belong to this sequence. - Keith Backman, Nov 09 2023

			9 is a term because 9-4 = 5 is prime, 9-2 = 7 is prime, 9+2 = 11 is prime and 9+4 = 13 is prime.

Klaus Brockhaus, Table of n, a(n) for n = 1..28388 (terms < 10^9).

Cf. A001359, A006512, A007530, A007811, A014561, A023200, A038800, A046132, A087680, A112540, A125855, A164385.

[ p+4: p in PrimesUpTo(100000) | IsPrime(p) and IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8) ]; // Klaus Brockhaus, Feb 09 2010

Select[Range[100000],AllTrue[#+{4,2,-2,-4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 30 2015 *)

is(n)=isprime(n-4) && isprime(n-2) && isprime(n+2) && isprime(n+4) \\ Charles R Greathouse IV, Sep 24 2015

from sympy import primerange
def aupto(limit):
    p, q, r, alst = 2, 3, 5, []
    for s in primerange(7, limit+5):
        if p+2 == q and p+6 == r and p+8 == s: alst.append(p+4)
        p, q, r = q, r, s
    return alst
print(aupto(10**5)) # Michael S. Branicky, Feb 03 2022

For n >= 2, a(n) = 15*A112540(n-1). - Michel Marcus, May 19 2014
From Jeppe Stig Nielsen, Feb 18 2020: (Start)
For n >= 2, a(n) = 30*A014561(n-1) + 15.
For n >= 2, a(n) = 10*A007811(n-1) + 5.
a(n) = A007530(n) + 4.
a(n) = A125855(n) + 5. (End)

Edited and extended beyond a(9) by Klaus Brockhaus, Feb 09 2010

A008471 Exactly 3 out of 10m+1, 10m+3, 10m+7, 10m+9 are primes.

Original entry on oeis.org

4, 7, 13, 22, 31, 43, 46, 61, 64, 85, 88, 103, 106, 109, 130, 142, 145, 160, 166, 169, 178, 199, 238, 268, 271, 316, 367, 376, 391, 400, 409, 415, 421, 451, 472, 478, 493, 523, 541, 544, 547, 550, 574

Offset: 1

Olivier Gérard

nonn

V. Raman, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A007811, A032352, A216292, A216293.

[n: n in [1..600] | #PrimesInInterval(10*n+1, 10*n+9) eq 3]; // Bruno Berselli, Sep 04 2012

m3Q[m_]:=Module[{c=10m},Count[{c+1,c+3,c+7,c+9},?PrimeQ]==3]; Select[ Range[ 600],m3Q] (* _Harvey P. Dale, Dec 24 2011 *)
Select[Range[600],Total[Boole[PrimeQ[10 #+{1,3,7,9}]]]==3&] (* Harvey P. Dale, Jul 27 2025 *)

A125855 Numbers k such that k+1, k+3, k+7 and k+9 are all primes.

Original entry on oeis.org

4, 10, 100, 190, 820, 1480, 1870, 2080, 3250, 3460, 5650, 9430, 13000, 15640, 15730, 16060, 18040, 18910, 19420, 21010, 22270, 25300, 31720, 34840, 43780, 51340, 55330, 62980, 67210, 69490, 72220, 77260, 79690, 81040, 82720, 88810, 97840

Offset: 1

Artur Jasinski, Dec 12 2006

nonn

It seems that, with the exception of 4, all terms are multiples of 10. - Emeric Deutsch, Dec 24 2006
In fact, all terms except 4 are congruent to 10 (mod 30). - Franklin T. Adams-Watters, Jun 05 2014
For n > 1: a(n) = 10*A007811(n-1). - Reinhard Zumkeller, Jul 18 2014 [Comment corrected by Jens Kruse Andersen, Jul 19 2014]

			For k = 10, the numbers 10 + 1 = 11, 10 + 3 = 13, 10 + 7 = 17, 10 + 9 = 19 are prime. - _Marius A. Burtea_, May 18 2019

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007530, A057015, A125779, A125780.

Cf. A010051, A245304 (subsequence), A007811.

a125855 n = a125855_list !! (n-1)
a125855_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [1, 1, 1, 1]) [1, 3, 7, 9]
-- Reinhard Zumkeller, Jul 18 2014

[n:n in [1..100000]| IsPrime(n+1) and IsPrime(n+3) and IsPrime(n+7) and IsPrime(n+9)]; // Marius A. Burtea, May 18 2019

a:=proc(n): if isprime(n+1)=true and isprime(n+3)=true and isprime(n+7)=true and isprime(n+9)=true then n else fi end: seq(a(n),n=1..500000); # Emeric Deutsch, Dec 24 2006

Do[If[(PrimeQ[x + 1]) && (PrimeQ[x + 3]) && (PrimeQ[x + 7]) && (PrimeQ[x + 9]), Print[x]], {x, 1, 10000}]
(* Second program *)
Select[Range[10^5], Times @@ Boole@ Map[PrimeQ, # + {1, 3, 7, 9}] == 1 &] (* Michael De Vlieger, Jun 12 2017 *)
Select[Range[100000],AllTrue[#+{1,3,7,9},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 02 2018 *)

is(n) = my(v=[1, 3, 7, 9]); for(t=1, #v, if(!ispseudoprime(n+v[t]), return(0))); 1 \\ Felix Fröhlich, May 18 2019

a(n) = A007530(n) - 1. - R. J. Mathar, Jun 14 2017

More terms from Emeric Deutsch, Dec 24 2006

A216292 Values of k such that there is exactly one prime between 10k and 10k + 9.

Original entry on oeis.org

9, 11, 12, 14, 18, 21, 24, 29, 30, 36, 39, 41, 42, 45, 47, 48, 55, 58, 63, 66, 68, 69, 71, 72, 74, 77, 78, 79, 80, 81, 83, 86, 87, 90, 92, 93, 95, 96, 98, 100, 102, 104, 105, 108, 111, 116, 117, 119, 120, 124, 125, 131, 137, 138, 139, 140, 144, 147, 151, 152

Offset: 1

V. Raman, Sep 03 2012

nonn

			36 is in the sequence because between 360 and 369 there is exactly one prime: 367. [_Bruno Berselli_, Sep 04 2012]

V. Raman, Table of n, a(n) for n = 1..10000

Cf. A007811, A032352, A078494, A216293.

[n: n in [1..200] | IsOne(#PrimesInInterval(10*n, 10*n+9))]; // Bruno Berselli, Sep 04 2012

t = {}; Do[ps = Select[Range[10*n, 10*n + 9], PrimeQ]; If[Length[ps] == 1, AppendTo[t, n]], {n, 0, 199}]; t (* T. D. Noe, Sep 03 2012 *)
Select[Range[200],PrimePi[10#+9]-PrimePi[10#]==1&] (* Harvey P. Dale, Feb 04 2015 *)

is(n)=isprime(10*n+1)+isprime(10*n+3)+isprime(10*n+7)+isprime(10*n+9)==1 \\ Charles R Greathouse IV, Sep 07 2012

from itertools import count, islice
from sympy import isprime
def A216292_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k: sum(int(isprime(10*k+i)) for i in (1,3,7,9)) == 1, count(max(1,startvalue)))
A216292_list = list(islice(A216292_gen(),30)) # Chai Wah Wu, Sep 23 2022

a(n) ~ 0.1 n log n. - Charles R Greathouse IV, Sep 07 2012

a(n) = floor(A078494(n) / 10). - Charles R Greathouse IV, Sep 07 2012

A216293 Values of k such that there are exactly two primes between 10k and 10k + 9.

Original entry on oeis.org

2, 3, 5, 6, 8, 15, 16, 17, 23, 25, 26, 27, 28, 33, 34, 35, 37, 38, 40, 44, 49, 50, 52, 54, 56, 57, 59, 60, 65, 67, 70, 73, 75, 76, 91, 94, 97, 99, 101, 110, 112, 115, 118, 121, 122, 123, 127, 128, 129, 132, 136, 143, 149, 154, 155, 157, 161, 162, 172, 174

Offset: 1

V. Raman, Sep 03 2012

nonn

			23 is in the sequence because between 230 and 239 there are exactly two primes: 233 and 239. [_Bruno Berselli_, Sep 04 2012]

V. Raman, Table of n, a(n) for n = 1..10000

Cf. A032352, A007811, A078494, A216292.

[n: n in [1..200] | #PrimesInInterval(10*n, 10*n+9) eq 2]; // Bruno Berselli, Sep 04 2012

t = {}; Do[ps = Select[Range[10*n, 10*n + 9], PrimeQ]; If[Length[ps] == 2, AppendTo[t, n]], {n, 0, 229}]; t (* T. D. Noe, Sep 03 2012 *)
Select[Range[200],Count[Range[10#,10#+9],?PrimeQ]==2&] (* _Harvey P. Dale, Jan 19 2017 *)

a(n) >> n log^2 n. - Charles R Greathouse IV, Sep 07 2012

cp's OEIS Frontend

A007530 Prime quadruples: numbers k such that k, k+2, k+6, k+8 are all prime.

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A032352 Numbers k such that there is no prime between 10*k and 10*k+9.

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A014561 Numbers k giving rise to prime quadruples (30k+11, 30k+13, 30k+17, 30k+19).

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A038800 Number of primes between 10n and 10n+9.

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PARI

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A067267 Numbers n such that n, 10*n+1, 10*n+3, 10*n+7 and 10*n+9 are all primes.

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Maple

Mathematica

A173037 Numbers k such that k-4, k-2, k+2 and k+4 are prime.

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Magma

A032352 Numbers k such that there is no prime between 10k and 10k+9.

A067267 Numbers n such that n, 10n+1, 10n+3, 10n+7 and 10n+9 are all primes.