A136162 -id:A136162 - 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

A275515 Table read by rows: list of prime triples of the form (p, p+2, p+6).

Original entry on oeis.org

5, 7, 11, 11, 13, 17, 17, 19, 23, 41, 43, 47, 101, 103, 107, 107, 109, 113, 191, 193, 197, 227, 229, 233, 311, 313, 317, 347, 349, 353, 461, 463, 467, 641, 643, 647, 821, 823, 827, 857, 859, 863, 881, 883, 887, 1091, 1093, 1097, 1277, 1279, 1283, 1301, 1303, 1307

Offset: 1

Arkadiusz Wesolowski, Jul 31 2016

A prime triple is a set of three prime numbers of the form (p, p+2, p+6) or (p, p+4, p+6).
Initial members p (other than 5) of prime triples of the form (p, p+2, p+6) are congruent to 11 or 17 (mod 30).
Also called prime triples of the first kind.

			The table starts:
5, 7, 11;
11, 13, 17;
17, 19, 23;
...

C. K. Caldwell, Top Twenty page, Triplet
Eric Weisstein's World of Mathematics, Prime Triplet
Wikipedia, Prime triple
Index entries for primes, gaps between

Cf. A022004, A077800, A136162, A270998, A275516.

&cat[[p, p+2, p+6]: p in PrimesUpTo(1301) | (p le 5 xor p mod 30 in {11, 17}) and IsPrime(p+2) and IsPrime(p+6)];

Prime@ Range[#, # + 2] &@ PrimePi@ Select[Prime@ Range@ 216, Times @@ Boole@ PrimeQ[# + {2, 6}] > 0 &] // Flatten (* Michael De Vlieger, Aug 02 2016 *)

a(3*n-2) = A022004(n).

A384298 Primes p such that p + 4, p + 12 and p + 16 are also primes.

Original entry on oeis.org

7, 67, 97, 487, 757, 1567, 1597, 2377, 3907, 7687, 8677, 12097, 12907, 13147, 14407, 14767, 15667, 16057, 19417, 21487, 31177, 38317, 43777, 52567, 57637, 58897, 65167, 65827, 67477, 67927, 74857, 81547, 90007, 90187, 93967, 94777, 95467, 95617, 102547, 111427, 112237, 114757, 123817, 129277

Offset: 1

Alexander Yutkin, May 25 2025

nonn

Initial members of prime quartets that correspond to the difference pattern [4, 8, 4].

			p=97: 97+4=101, 97+12=109, 97+16=113 —> prime quartet: (97, 101, 109, 113).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A001223.

Cf. A136162 [2, 4, 2], A052378 [4, 2, 4], A382810 [6, 4, 6].

q:= n-> andmap(i-> isprime(n+4*i), [0,1,3,4]):
select(q, [7+30*i$i=0..4309])[];  # Alois P. Heinz, May 29 2025

Select[Prime[Range[12099]],AllTrue[#+{4,12,16},PrimeQ]&] (* James C. McMahon, May 29 2025 *)

a(n) == 7 (mod 30).

A222960 Numbers n such that 2*n + {3, 5, 9, 11} are all primes.

Original entry on oeis.org

1, 4, 49, 94, 409, 739, 934, 1039, 1624, 1729, 2824, 4714, 6499, 7819, 7864, 8029, 9019, 9454, 9709, 10504, 11134, 12649, 15859, 17419, 21889, 25669, 27664, 31489, 33604, 34744, 36109, 38629, 39844, 40519, 41359, 44404

Offset: 1

Vincenzo Librandi, Mar 13 2013

After 1, a(n) is congruent to 4 or 9 (mod 10), this means that a(n) is of the form 5k-1. Clearly, each term is not divisible by 3 and 11.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007530, A136162.

[n: n in [1..50000] | forall{2*n+k: k in [3,5,9,11] | IsPrime(2*n+k)}];

Select[Range[50000], Union[PrimeQ[2 # + {3, 5, 9, 11}]]=={True}&]

a(n) = (A007530(n)-3)/2. [Michael B. Porter, Mar 21 2013]

A222961 Numbers n such that 2*n + {3, 5, 9, 11, 15, 21} are all primes.

Original entry on oeis.org

1, 4, 739, 82849, 163069, 330544, 511249, 534349, 623179, 730699, 958864, 1529434, 2077954, 2109139, 2763499, 4172074, 5067199, 5882524, 5950249, 6532159, 7860409, 7880044, 9252184, 9368419, 9564769, 9605719, 10024984

Offset: 1

Vincenzo Librandi, Mar 13 2013

nonn

After 1, a(n) is congruent to 4 or 9 (mod 10), this means that a(n) is of the form 5k-1. Clearly, each term is not divisible by 3, 7 and 11.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A136162.

[n: n in [1..11000000] | forall{2*n+k: k in [3, 5, 9, 11, 15, 21] | IsPrime(2*n+k)}];

Select[Range[7000000], Union[PrimeQ[2 # +{3, 5, 9, 11, 15, 21}]]=={True}&]

A384299 Primes p such that p + 8, p + 12 and p + 20 are also primes.

Original entry on oeis.org

11, 59, 89, 389, 479, 1439, 1559, 1601, 2531, 2699, 3209, 3449, 3911, 5639, 5849, 7529, 8081, 8669, 10091, 12269, 12401, 12899, 13151, 14411, 14759, 17021, 19421, 21011, 21851, 22271, 23189, 25931, 26099, 28649, 28859, 31139, 31469, 33191, 33569, 36551, 39659, 40751, 42689, 43391, 43781, 44111

Offset: 1

Alexander Yutkin, May 25 2025

nonn

Initial members of prime quartets that correspond to the difference pattern [8, 4, 8].

			p=89: 89+8=97, 89+12=101, 89+20=109 —> prime quartet: (89, 97, 101, 109).

Cf. A000040, A001223.

Cf. A136162 [2, 4, 2], A052378 [4, 2, 4], A382810 [6, 4, 6].

q:= n-> andmap(i-> isprime(n+4*i), [0,2,3,5]):
select(q, [5+6*i$i=1..7351])[];  # Alois P. Heinz, May 29 2025

Select[Prime[Range[4591]],AllTrue[#+{8,12,20},PrimeQ]&] (* James C. McMahon, May 29 2025 *)

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

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A275515 Table read by rows: list of prime triples of the form (p, p+2, p+6).

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A384298 Primes p such that p + 4, p + 12 and p + 16 are also primes.

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A222960 Numbers n such that 2*n + {3, 5, 9, 11} are all primes.

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A222961 Numbers n such that 2*n + {3, 5, 9, 11, 15, 21} are all primes.

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Magma

Mathematica

A384299 Primes p such that p + 8, p + 12 and p + 20 are also primes.

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Examples

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Programs

Maple

Mathematica