A007530 -id:A007530 - cp's OEIS frontend

A113404 Record gaps between prime quadruplets.

6, 90, 630, 660, 1170, 2190, 3780, 6420, 8940, 9030, 13260, 16470, 24150, 28800, 29610, 39990, 56580, 56910, 71610, 83460, 94530, 114450, 157830, 159060, 171180, 177360, 190500, 197910, 268050, 315840, 395520, 435240, 440910, 513570, 536010, 539310, 557340, 635130

Offset: 1

Bernardo Boncompagni, Oct 28 2005

nonn

Prime quadruplets (p, p+2, p+6, p+8) are densest permissible constellations of 4 primes (A007530). By the Hardy-Littlewood k-tuple conjecture, average gaps between prime k-tuples are O(log^k(p)), with k=4 for quadruplets. If a gap is larger than all preceding gaps, we call it a maximal gap, or a record gap. Maximal gaps may be significantly larger than average gaps. This sequence suggests that maximal gaps between prime quadruplets are O(log^5(p)). - Alexei Kourbatov, Jan 04 2012

			The first prime quadruplets are (5,7,11,13) and (11,13,17,19), so a(1)=11-5=6. The next quadruplet is (101,103,107,109), so a(2)=101-11=90. The following quadruplet is 191,193,197,199 so 90 remains the record and no terms are added.

Alexei Kourbatov, Table of n, a(n) for n = 1..71
T. Forbes, Norman Luhn Prime k-tuplets
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
Alexei Kourbatov, Maximal gaps between prime k-tuples (graphs, more terms)
A. Kourbatov, Maximal gaps between prime k-tuples: a statistical approach, arXiv preprint arXiv:1301.2242 [math.NT], 2013. - From _N. J. A. Sloane_, Feb 09 2013
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, The distribution of maximal prime gaps in Cramer's probabilistic model of primes, arXiv preprint arXiv:1401.6959 [math.NT], 2014.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.
Eric Weisstein's World of Mathematics, Hardy-Littlewood Constants.
Eric Weisstein's World of Mathematics, Prime Constellation.

A229907 lists initial primes in quadruplets preceding the maximal gaps. A113403 lists the corresponding primes at the end of the maximal gaps. Cf. A008407, A007530.

DeleteDuplicates[Differences[#[[4]]&/@Select[Partition[Prime[Range[10^7]],4,1],Differences[#] == {2,4,2}&]],GreaterEqual] (* The program generates the first 29 terms of the sequence. *) (* Harvey P. Dale, Aug 04 2024 *)

From Alexei Kourbatov, Jan 04 2012: (Start)
(1) Upper bound: gaps between prime quadruplets (p, p+2, p+6, p+8) are smaller than 0.241*(log p)^5, where p is the prime at the end of the gap.
(2) Estimate for the actual size of the maximal gap that ends at p: maximal gap = a*(log(p/a)-0.55), where a = 0.241*(log p)^4 is the average gap between quadruplets near p, as predicted by the Hardy-Littlewood k-tuple conjecture.
Formulas (1) and (2) are asymptotically equal as p tends to infinity. However, (1) yields values greater than all known gaps, while (2) yields "good guesses" that may be either above or below the actual size of known maximal gaps.
Both formulas (1) and (2) are derived from the Hardy-Littlewood k-tuple conjecture via probability-based heuristics relating the expected maximal gap size to the average gap. Neither of the formulas has a rigorous proof (the k-tuple conjecture itself has no formal proof either). In both formulas, the constant ~0.241 is reciprocal to the Hardy-Littlewood 4-tuple constant 4.15118... (End)

Terms 159060 to 635130 added by Alexei Kourbatov, Jan 04 2012

A049437 Primes p such that p+2 and p+8 are also primes but p+6 is not.

Original entry on oeis.org

3, 29, 59, 71, 149, 269, 431, 569, 599, 1031, 1061, 1229, 1289, 1319, 1451, 1619, 2129, 2339, 2381, 2549, 2711, 2789, 3299, 3539, 4019, 4049, 4091, 4649, 4721, 5099, 5441, 5519, 5639, 5741, 5849, 6269, 6359, 6569, 6701, 6959, 7211

Offset: 1

Labos Elemer

nonn

p+4 is not prime here except for p=3.

			p=29 is the smallest prime so that p, p+2 and p+8 are consecutive primes.

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from R. J. Mathar)

Cf. A007530, A023202, A031926, A046134, A046138, A049436, A049438, A046138.

Subsequence of A001359. - R. J. Mathar, Feb 10 2013

[p: p in PrimesUpTo(8000)| IsPrime(p+2) and IsPrime(p+8) and not IsPrime(p+6) ] // Vincenzo Librandi, Jan 28 2011

select(p -> isprime(p) and isprime(p+2) and isprime(p+8) and not isprime(p+6), [3, seq(i,i=5..10000, 6)]); # Robert Israel, Nov 20 2017

{3}~Join~Select[Partition[Prime@ Range[10^3], 3, 1], Differences@ # == {2, 6} &][[All, 1]] (* Michael De Vlieger, Nov 20 2017 *)

lista(nn) = forprime(p=3, nn, if(isprime(p+2) && isprime(p+8) && !isprime(p+6), print1(p, ", "))) \\ Iain Fox, Nov 20 2017

A049438 p, p+6 and p+8 are all primes (A046138) but p+2 is not.

Original entry on oeis.org

23, 53, 131, 173, 233, 263, 563, 593, 653, 1013, 1223, 1283, 1601, 1613, 2333, 2543, 2963, 3323, 3533, 3761, 3911, 3923, 4013, 4211, 4253, 4643, 4793, 5003, 5273, 5471, 5843, 5861, 6263, 6353, 6563, 6653, 6863, 7121, 7451, 7481, 7541, 7583

Offset: 1

Labos Elemer

R. J. Mathar, Table of n, a(n) for n = 1..1000

Subsequence of A031924. - R. J. Mathar, Jun 15 2013

Cf. A007530, A023202, A031926, A046134, A046138, A049436, A049437.

Select[Prime@ Range[10^3], MatchQ[Boole@ PrimeQ@ {# + 2, # + 6, # + 8}, {0, 1, 1}] &] (* Michael De Vlieger, Feb 05 2017 *)

isok(p) = isprime(p) && !isprime(p+2) && isprime(p+6) && isprime(p+8); \\ Michel Marcus, Dec 13 2013

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

A053778 First of four consecutive primes that comprise two sets of twin primes.

Original entry on oeis.org

5, 11, 101, 137, 179, 191, 419, 809, 821, 1019, 1049, 1481, 1871, 1931, 2081, 2111, 2969, 3251, 3359, 3371, 3461, 4217, 4229, 4259, 5009, 5651, 5867, 6689, 6761, 6779, 6947, 7331, 7547, 8219, 8969, 9419, 9431, 9437, 10007, 11057, 11159, 11699, 12239

Offset: 1

Labos Elemer, Mar 24 2000

nonn

These twins are not necessarily at the minimal distance as in A007530 (which is a subsequence).

			These primes initiate consecutive p quadruples as follows: [p,p+2,p+6k,p+6k+2]. For 6k=6,12,18,24,30,36,54 such a p =5,137,1931,9437,2968, 20441 and 48677 resp. Such a quadruple is [48677,48679,48731,48733], with [2,52,2] difference pattern.

M. F. Hasler, Table of n, a(n) for n = 1..5274.
Eric Weisstein's World of Mathematics, Prime Quadruplet

Cf. A001223, A001359, A007530, A048614.

Transpose[Select[Partition[Prime[Range[1500]],4,1],#[[4]]-#[[3]]== #[[2]]-#[[1]]== 2&]][[1]] (* Harvey P. Dale, Jul 07 2011 *)

forprime( p=1,10^5, isprime(p+2) || next; isprime(nextprime(p+4)+2) && print1(p","))

nextA053778(p)=until( isprime(nextprime(p+1)+2), until( p+2==p=nextprime(p+1),)); p-2

p=0; A053778=vector(100,i, p=nextA053778(p+1))

A001359 primes for which A048614 is zero. Lesser of 2-twin primes after which the consecutive prime difference pattern (of A001223) is [2, 6k-2, 2] for some k.

Edited by N. J. A. Sloane, Apr 13 2008, at the suggestion of M. F. Hasler.

A059925 Numbers n such that {n, n+2, n+6, n+8, n+30, n+32, n+36, n+38} are all prime.

Original entry on oeis.org

1006301, 2594951, 3919211, 9600551, 10531061, 108816311, 131445701, 152370731, 157131641, 179028761, 211950251, 255352211, 267587861, 557458631, 685124351, 724491371, 821357651, 871411361, 1030262081, 1103104361, 1282160021, 1381201271, 1427698631, 1432379951, 1443994001

Offset: 1

Martin Raab, Mar 03 2001

nonn

Each term is the initial member of two prime quadruples (A007530) with the smallest possible difference of 30.

Jud McCranie and Sebastian Petzelberger, Table of n, a(n) for n = 1..10000 (first 1238 terms from Jud McCranie)
J. Brüggemann, The twins of prime quadruples up to 10^17 [71 MB].
D. La Pierre Ballard, Prime Number Quadruplets 30 Apart

Cf. A007530, A256842.

Select[Prime[Range[5582*10^4]],AllTrue[#+{2,6,8,30,32,36,38},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2019 *)

is(n)=my(v=[0,2,6,8,30,32,36,38]);for(i=1,8, if(!isprime(n+v[i]), return(0)));1 \\ Charles R Greathouse IV, Jun 18 2013

a(n) = 2 (mod 21). - Hugo Pfoertner, Dec 29 2024

For clarity, replaced definition by a comment from Charles R Greathouse IV. - N. J. A. Sloane, Nov 26 2020

A136162 List of prime quadruplets {p, p+2, p+6, p+8}.

Original entry on oeis.org

5, 7, 11, 13, 11, 13, 17, 19, 101, 103, 107, 109, 191, 193, 197, 199, 821, 823, 827, 829, 1481, 1483, 1487, 1489, 1871, 1873, 1877, 1879, 2081, 2083, 2087, 2089, 3251, 3253, 3257, 3259, 3461, 3463, 3467, 3469, 5651, 5653, 5657, 5659, 9431, 9433, 9437

Offset: 1

Harry J. Smith, Dec 17 2007

{11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12*k^2-1) and 3*(12*k^2+1) respectively (k is an integer). Conjecture: {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {q*(nextprime(q))-4, q*( nextprime(q))-2, q*( nextprime(q))+2, q*( nextprime(q))+4} where q is a prime (for prime q = 3). - Jaroslav Krizek, Jul 07 2017

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Quadruplet

Cf. A007530 (1st quadrisection).

Map[Prime[Range @@ #] &, MapAt[# + 1 &, SequencePosition[Differences@ Prime@ Range@ 1200, {2, 4, 2}], {All, -1}]] // Flatten (* Michael De Vlieger, Jul 11 2017 *)

{forprime(p1=0,70000,p2=p1+2;if(!isprime(p2),next;);p3=p1+6;if(!isprime(p3),next;);p4=p1+8;if(!isprime(p4),next;);print1(p1,",",p2,",",p3,",",p4,","))}

q=[0,0,0,0];i=0;forprime(p=5,1e4,(q[i%4+1]=p)==8+q[i++%4+1]&&print1(vecsort(q)","))  \\ M. F. Hasler, Apr 20 2013

[a(4n-3),a(4n-2),a(4n-1),a(4n)] = A007530(n) + [0,2,6,8], for all n>0. - M. F. Hasler, Apr 20 2013

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

A090258 Last term of prime quadruples.

Original entry on oeis.org

13, 19, 109, 199, 829, 1489, 1879, 2089, 3259, 3469, 5659, 9439, 13009, 15649, 15739, 16069, 18049, 18919, 19429, 21019, 22279, 25309, 31729, 34849, 43789, 51349, 55339, 62989, 67219, 69499, 72229, 77269, 79699, 81049, 82729, 88819, 97849

Offset: 1

Cino Hilliard, Jan 24 2004

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Equals A064974 - 1.

Equals A007530 + 8.

A090258 = Select[Range[9, 10^5 - 1, 2], Union[PrimeQ[# - {0, 2, 6, 8}]] == {True} &] (* Alonso del Arte, Aug 12 2012 *)
Transpose[Select[Partition[Prime[Range[9500]],4,1],Differences[#]=={2,4,2}&]] [[4]] (* Harvey P. Dale, Nov 11 2013 *)

quintpr(n) = { s=0; forprime(p=5,n, if(isprime(p+2) && isprime(p+6) && isprime(p+8), print1(p+8","); ) ); print(); print(s+.0) }

A047299 Numbers that are congruent to {0, 1, 3, 4, 6} mod 7.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 69, 70, 71, 73, 74, 76, 77, 78

Offset: 1

N. J. A. Sloane

Nonnegative m such that floor(k*m^2/7) = k*floor(m^2/7), where k = 2 or 3. [Bruno Berselli, Dec 03 2015]

For k > 1 (A007530(k+1) - A007530(k))/30 is a term in this sequence. - Hugo Pfoertner, May 29 2020

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A007530 (prime quadruples).

[Floor((7*n-5)/5): n in [1..100]]; // Zaki Khandaker, Jun 21 2015

a[n_]:=Floor[(7n-5)/5]; Table[a[i],{i,1,30}]; (* Lorenz H. Menke, Jr., Jun 19 2013 *)
LinearRecurrence[{1,0,0,0,1,-1},{0,1,3,4,6,7},80] (* Harvey P. Dale, Jan 24 2025 *)

a(n)=(7*n-5)\5 \\ Charles R Greathouse IV, Jun 19 2013

G.f.: x^2*(1+2*x+x^2+2*x^3+x^4) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011

a(n) = floor((7n-5)/5). - Lorenz H. Menke, Jr., Jun 19 2013

Formula and programs adapted to offset 1 by Michel Marcus, May 30 2020

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A113404 Record gaps between prime quadruplets.

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A049437 Primes p such that p+2 and p+8 are also primes but p+6 is not.

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A049438 p, p+6 and p+8 are all primes (A046138) but p+2 is not.

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

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A053778 First of four consecutive primes that comprise two sets of twin primes.

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A059925 Numbers n such that {n, n+2, n+6, n+8, n+30, n+32, n+36, n+38} are all prime.

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A136162 List of prime quadruplets {p, p+2, p+6, p+8}.

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