A014561 -id:A014561 - 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

A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

Original entry on oeis.org

1, 10, 19, 82, 148, 187, 208, 325, 346, 565, 943, 1300, 1564, 1573, 1606, 1804, 1891, 1942, 2101, 2227, 2530, 3172, 3484, 4378, 5134, 5533, 6298, 6721, 6949, 7222, 7726, 7969, 8104, 8272, 8881, 9784, 9913, 10111, 10984, 11653, 11929, 12220, 13546, 14416, 15727

Offset: 1

N. J. A. Sloane and J. H. Conway, Mar 15 1996

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A024912, A102338, A102342, A102700, A007530, A014561, A008471, A032352, A216292, A216293, A125855.

Cf. A010051, A245304, A245305.

a007811 n = a007811_list !! (n-1)
a007811_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [10, 10, 10, 10]) [1, 3, 7, 9]
-- Reinhard Zumkeller, Jul 18 2014

[n: n in [0..10000] | forall{10*n+r: r in [1,3,7,9] | IsPrime(10*n+r)}]; // Bruno Berselli, Sep 04 2012

for n from 1 to 10000 do m := 10*n: if isprime(m+1) and isprime(m+3) and isprime(m+7) and isprime(m+9) then print(n); fi: od: quit

Select[ Range[ 1, 10000, 3 ], PrimeQ[ 10*#+1 ] && PrimeQ[ 10*#+3 ] && PrimeQ[ 10*#+7 ] && PrimeQ[ 10*#+9 ]& ]
Select[Range[15000], And @@ PrimeQ /@ ({1, 3, 7, 9} + 10#) &] (* Ray Chandler, Jan 12 2007 *)

p=2;q=3;r=5;forprime(s=7,1e5,if(s-p==8 && r-p==6 && q-p==2 && p%10==1, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Mar 21 2013

use ntheory ":all"; my @s = map { ($-1)/10 } sieve_prime_cluster(10,1e9, 2,6,8); say for @s; # _Dana Jacobsen, May 04 2017

a(n) = 3*A014561(n) + 1. - Zak Seidov, Sep 21 2009

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

A182279 Numbers n such that 30n+{11, 13, 17, 19, 23} are 5 consecutive primes.

Original entry on oeis.org

0, 3, 49, 535, 647, 700, 742, 1459, 1844, 4805, 5523, 5561, 6524, 6727, 7511, 8253, 8960, 10871, 11599, 12040, 13258, 15505, 17293, 17881, 21115, 21126, 22036, 25606, 26526, 27657, 28598, 29200, 31951, 33628, 34083, 35465, 35623, 36375, 39084, 39119, 40362

Offset: 1

Zak Seidov, Apr 23 2012

nonn

All terms are congruent to {0, 3} mod 7.

			a(3) = 49 = A014561(5), a(4) = 535 = A014561(15).

Zak Seidov, Table of n, a(n) for n = 1..1000

Subsequence of A014561.

Cf. A182282.

isok(n) = isprime(p=30*n+11) && ((q=nextprime(p+1))==(30*n+13)) && ((r=nextprime(q+1))==(30*n+17)) && ((s=nextprime(r+1))==(30*n+19)) && ((t=nextprime(s+1))==(30*n+23)); \\ Michel Marcus, Oct 19 2013

A182282 Numbers n such that 210*n+{11, 13, 17, 19, 23, 29} are 6 consecutive primes.

Original entry on oeis.org

0, 7, 789, 1553, 3148, 4869, 5089, 5935, 6959, 9132, 14566, 19790, 20087, 26319, 39734, 48259, 56024, 56669, 62211, 74861, 75048, 88116, 89223, 91093, 91483, 95476, 100172, 113159, 122183, 130160, 152125, 160557, 163247, 164501, 167811, 176585, 187771, 189250

Offset: 1

Zak Seidov, Apr 23 2012

nonn

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A014561, A182279.

a(3)=789=A182279(11)/7, a(4)=1553=A182279(18)/7.

Select[Range[0,200000],AllTrue[210#+{11,13,17,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 28 2016 *)

A182387 Numbers k such that 210*k+{11,13,17,19,23,29,31} are 7 consecutive primes.

Original entry on oeis.org

0, 789, 5089, 56669, 75048, 88116, 100172, 122183, 187771, 214298, 322935, 413420, 445838, 593886, 648863, 667224, 736358, 772329, 868588, 888020, 890616, 907211, 945016, 1052954, 1078331, 1106177, 1146724, 1223888, 1432230, 1452437, 1458355, 1509878, 1535216

Offset: 1

Zak Seidov, Apr 27 2012

nonn

Subsequence of A182282: a(2)=789=A182282(3), a(3)=5089=A182282(7), etc.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A014561, A182279, A182282.

Select[Range[0,1600000],And@@PrimeQ[210 #+{11,13,17,19,23,29,31}]&] (* Harvey P. Dale, Mar 04 2013 *)

A061671 Numbers n such that { x +- 2^k : 0 < k < 4 } are primes, where x = 210*n - 105.

Original entry on oeis.org

1, 77, 93, 209, 5197, 7695, 9307, 13442, 13524, 15445, 16192, 28600, 30970, 34228, 36388, 38391, 41625, 50127, 52795, 55546, 69146, 70538, 70642, 70747, 76314, 76642, 90079, 91416, 93496, 94288, 95773, 96415, 101530, 104049, 107559, 118031

Offset: 1

Frank Ellermann, Jun 16 2001

nonn

This sequence does not include the sextet (7,11,13,17,19,23). It is a proper subset of A014561 in a certain sense.

			16057, 16061, 16063, 16067, 16069, 16073 are prime and (16065+105)/210= 77= a(2).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, conjectures following th. 5

Harvey P. Dale, Table of n, a(n) for n = 1..900
F. Ellermann, Illustration for A002110, A005867, A038110, A060753

210 = 7*5*3*2 = A002110(4), cf. A014561.

Select[Range[1, 1000000], Union[PrimeQ[(210*# - 105) + {-8, -4, -2, 2, 4, 8}]] == {True} &]
Select[Range[120000],AllTrue[210#-105+{-8,-4,-2,2,4,8},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 05 2019 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 20 2001 and from Frank Ellermann, Nov 26 2001. Mathematica script from Peter Bertok (peter(AT)bertok.com), Nov 27 2001.

A066081 a(n) = smallest m such that m+2^j and m-2^j are prime for all 0 < j <= n.

Original entry on oeis.org

5, 9, 15, 50943795, 40874929095, 616517522595975, 93487500801880185, 64606701602327559675

Offset: 1

Frank Ellermann, Dec 03 2001

Is this sequence infinite?

			9-4, 9-2, 9+2, 9+4 are prime, but not 5+4 = 7+2, therefore a(2) = 9.

Felice Russo, Prime puzzle 167.
Marek Wolf, Conjectures on the gaps between consecutive primes

Prime quadruples: A014561, sextets: A061671, octets: A066082.

a(5) and a(6) from Don Reble, Dec 07 2001

a(7) from Jim Fougeron (Feb 07) confirmed by Phil Carmody, who also found a(8) (Feb 14 2002).

A112540 Numbers m such that (15m-4, 15m-2, 15m+2, 15m+4) is a prime quadruple.

Original entry on oeis.org

1, 7, 13, 55, 99, 125, 139, 217, 231, 377, 629, 867, 1043, 1049, 1071, 1203, 1261, 1295, 1401, 1485, 1687, 2115, 2323, 2919, 3423, 3689, 4199, 4481, 4633, 4815, 5151, 5313, 5403, 5515, 5921, 6523, 6609, 6741, 7323, 7769, 7953, 8147, 9031, 9611, 10485, 11047

Offset: 1

Karsten Meyer, Dec 16 2005

Also (4p + 16)/60 such that (p, p+2, p+6 and p+8) is a prime quadruple for p >= 11. - Michel Lagneau, Jul 02 2012
The density of these four-prime groups is approximately equal to (log x)^-3.45 (but not (log x)^-4). - Xueshi Gao, Jun 01 2014
All of the terms of this sequence are either 1, 7 or 13 modulo 14. - Rodolfo Ruiz-Huidobro, Dec 27 2019
From Eric Snyder, Jun 23 2021: (Start)
Building on the comment of R. Ruiz-Huidobro above, all terms of the sequence are congruent to one of {-1, 0 ,1} (mod 7). The appearance of mod 14 stems from the fact that all entries in this list must be odd. Equivalently, a(n) cannot be +- 2 or +- 3 (mod 7). This can be generalized for all larger primes:
All primes p >= 7 can be expressed as 15k +- q in a least absolute residue system, with q in {2, 4} if k is odd, and q in {1,7} if k is even.
For all primes 15k +- q >= 7, four residues +-r, +-s (mod p) exist such that, if for any p >= 7, [(m == +- r (mod p) or m == +- s (mod p)), and (m != k)], then m is not in this sequence. For the different values of p = 15k +- q, the values of +-r and +-s are as follows:
    For p = 15k +- 1 (k even), r = +- 2k, s = +- 4k
    For p = 15k +- 2 (k odd), r = +- k, s = +- 2k
    For p = 15k + 4 (k odd), r = +- k, s = +- (7k + 2)
    For p = 15k - 4 (k odd), r = +- k, s = +- (7k - 2)
    For p = 15k + 7 (k even), r = +- (4k + 2), s = +- (8k + 4)
    For p = 15k - 7 (k even), r = +- (4k - 2), s = +- (8k - 4)
These can be used to create an Eratosthenes-like sieve for the prime decades. (End)

			m = 7 yields the quadruple (15*7 - 4 = 101, 15*7 - 2 = 103, 15*7 + 2 = 107, 15*7 + 4 = 109), so 7 is a term of the sequence.

Dana Jacobsen, Table of n, a(n) for n = 1..10000
Eric Snyder, An Eratosthenean Sieve for the Prime Decades

Cf. A007530, A014561.

[n: n in [0..2*10^4] | IsPrime(15*n-4) and IsPrime(15*n-2) and IsPrime(15*n+2) and IsPrime(15*n+4)]; // Vincenzo Librandi, Dec 28 2015

A112540:=n->`if`(isprime(15*n-4) and isprime(15*n-2) and isprime(15*n+2) and isprime(15*n+4),n,NULL); seq(A112540(n), n=1..20000); # Wesley Ivan Hurt, Jul 26 2014

Select[Range[6610], PrimeQ[15# - 4] && PrimeQ[15# - 2] && PrimeQ[15# + 2] && PrimeQ[15# + 4]&] (* T. D. Noe, Nov 16 2006 *)

for(n=1, 1e4, if(isprime(15*n-4) && isprime(15*n-2) && isprime(15*n+2) && isprime(15*n+4), print1(n, ", "))) \\ Felix Fröhlich, Jul 26 2014

use ntheory ":all"; say for map { (4*$+16)/60 } sieve_prime_cluster(11,15*10000, 2,6,8); # _Dana Jacobsen, Dec 15 2015

from sympy import isprime
def ok(m): return all(isprime(15*m+k) for k in [-4, -2, 2, 4])
print(list(filter(ok, range(11111)))) # Michael S. Branicky, Jun 24 2021

Corrected by T. D. Noe, Nov 16 2006

A066082 Prime octets: numbers k such that 210*k - 105 +- 2^j are prime for all 1 <= j <= 4.

Original entry on oeis.org

242590, 1175444, 2416288, 2583146, 2596049, 2796151, 4953911, 5574794, 6127655, 6396209, 6460877, 6625438, 8521234, 11025856, 11352491, 15482298, 16228703, 18861024, 19048003, 20043534, 22835193, 31519781, 34399756

Offset: 1

Frank Ellermann, Dec 03 2001

nonn

			a(1)=242590 because 210*a(1) - 105 = 50943795 and 50943795 -2, - 4, - 8, - 16, + 2, + 4, + 8 and + 16 are all prime.

Harry J. Smith, Table of n, a(n) for n = 1..100

A proper subset of A061671 (prime sextets). Cf. prime quadruples A014561. See also A066081.

With[{c=2^Range[4]},Select[Range[35*10^6],AllTrue[Flatten[210#-105+ {c,-c}], PrimeQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 10 2015 *)

{ n=0; for (m=1, 10^12, b=210*m - 105; c=0; for (j=1, 4, if (isprime(b - 2^j) , c++, break)); if (c<4, next); for (j=1, 4, if (isprime(b + 2^j) , c++, break)); if (c == 8, write("b066082.txt", n++, " ", m); if (n==100, return)) ) } \\ Harry J. Smith, Nov 11 2009

More terms from Don Reble, Dec 07 2001

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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A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

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

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A182279 Numbers n such that 30n+{11, 13, 17, 19, 23} are 5 consecutive primes.

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PARI

A182282 Numbers n such that 210*n+{11, 13, 17, 19, 23, 29} are 6 consecutive primes.

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Maple

Mathematica

A182387 Numbers k such that 210*k+{11,13,17,19,23,29,31} are 7 consecutive primes.

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Mathematica

A061671 Numbers n such that { x +- 2^k : 0 < k < 4 } are primes, where x = 210*n - 105.

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