A100423 -id:A100423 - cp's OEIS frontend

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

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

A100418 Numbers k such that 30*k + {1,11,13,17,19,23,29} are all prime.

Original entry on oeis.org

49, 34083, 41545, 48713, 140609, 524027, 616812, 855281, 1314397, 1324750, 1636152, 2281293, 2927134, 3401412, 3605413, 4989341, 5212221, 5284979, 5406303, 5645269, 6141254, 6342728, 7231434, 7347697, 7637329, 8027068, 8161657, 8372756, 8392776, 8567216, 8986096, 9145563

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004

Values are 0 mod 7.
From Peter Munn, Sep 06 2023: (Start)
In each case, the 7 primes are necessarily consecutive.
As A065706 demonstrates, many intervals of 27 integers contain 8 primes, but only A364678(30) = 7 primes can occur between adjacent positive multiples of 30. This is because there are 8 values {1,7,11,13,17,19,23,29} coprime to 30, but they cover every residue class modulo 7, which means at least one of 30*k + {1,7,11,13,17,19,23,29} is divisible by 7.
1 and 29 are in the same residue class, but if we remove any of the other coprime integers there is a class that is not represented in the set. For this sequence, we remove 7, so when k is a multiple of 7, none of 30*k + {1,11,13,17,19,23,29} is a multiple of 2, 3, 5 or 7 and the set can potentially be 7 consecutive primes.
The sequences for the other appropriate subsets of 7 coprime values are A100419-A100423.
(End)

David A. Corneth, Table of n, a(n) for n = 1..10309

Cf. A005776, A007775, A056956, A065706, A076205, A100419-A100423, A364678.

[ n: n in [0..70000000 by 7] | forall{ q: q in [1, 11, 13, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[803*10^4],AllTrue[30#+{1,11,13,17,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 11 2019 *)

{pav7(mx)= local(wp=[1,11,13,17,19,23,29],v=[],i,j,m); for(k=1,mx, i=k*30;j=1;m=1;while(m&&(j<8),m=isprime(i+wp[j]);j+=1);if(m,v=concat(v,k))); return(v)}

Edited by Don Reble, Nov 17 2005

A076205 Numbers n such that 30*n+{1,7,11,13,17,19,23,29} are all composite.

Original entry on oeis.org

360, 523, 654, 941, 1020, 1047, 1064, 1136, 1188, 1213, 1264, 1280, 1343, 1355, 1445, 1477, 1515, 1526, 1530, 1533, 1582, 1623, 1652, 1693, 1842, 1900, 1960, 2018, 2039, 2129, 2208, 2280, 2309, 2332, 2406, 2413, 2440, 2499, 2539, 2622, 2633, 2650, 2657

Offset: 1

Donald S. McDonald, Nov 02 2002

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 141.

Klaus Brockhaus, Table of n, a(n) for n = 1..6829 (terms < 100000)

Cf. A005776, A007775, A100418-A100423.

[ n: n in [0..3000] | forall{ q: q in [1, 7, 11, 13, 17, 19, 23, 29] | not IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[3000],AllTrue[30#+{1,7,11,13,17,19,23,29},CompositeQ]&] (* Harvey P. Dale, Jan 06 2022 *)

{cav(mx)= local(wp=[1,7,11,13,17,19,23,29],v=[],i,j,m); for(k=1,mx, i=k*30;j=1;m=1;while(m&&(j<9),m=!isprime(i+wp[j]);j+=1);if(m,v=concat(v,k))); return(v)}

More terms from Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004
Edited by Don Reble, Nov 17 2005
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar

A100419 Numbers k such that 30*k+{1,7,13,17,19,23,29} are all prime.

Original entry on oeis.org

89, 6627, 18674, 223949, 229269, 240007, 267356, 606681, 638454, 771496, 951060, 1068030, 1150693, 1254839, 1688923, 1920084, 2413577, 2433289, 2649414, 3053398, 3080572, 3337444, 3586658, 3604256, 3830335, 4137166

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004

nonn

Values are 5 mod 7.

In each case, the 7 primes are necessarily consecutive. See the comment in A100418. - Peter Munn, Sep 06 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..2020 from Robert Israel)

Cf. A005776, A007775, A076205, A100418, A100420-A100423.

[ n: n in [5..70000000 by 7] | forall{ q: q in [1, 7, 13, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

filter:= proc(n) local j; andmap(isprime, [seq(30*n+j,j=[1,7,13,17,19,23,29])]) end proc:
select(filter, [seq(i,i=5..5*10^6,7)]); # Robert Israel, Nov 04 2024

Select[Range[42*10^5],AllTrue[30#+{1,7,13,17,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 10 2018 *)

Edited by Don Reble, Nov 17 2005

A100420 Numbers n such that 30*n+{1,7,11,17,19,23,29} are all prime.

Original entry on oeis.org

22621, 103205, 149125, 237794, 288467, 321451, 364921, 373370, 404002, 851099, 985933, 1106235, 1594044, 1696874, 1780265, 1824421, 1851756, 2249881, 3112939, 3257538, 3397608, 3601651, 3747356, 4347340, 4710990, 4886284

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004

Values are 4 mod 7.

In each case, the 7 primes are necessarily consecutive. See the comment in A100418. - Peter Munn, Sep 06 2023

Cf. A005776, A007775, A076205, A100418, A100419, A100421-A100423.

[ n: n in [4..70000000 by 7] | forall{ q: q in [1, 7, 11, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[5000000],And@@PrimeQ[30 #+{1,7,11,17,19,23,29}]&]  (* Harvey P. Dale, Mar 06 2011 *)

Edited by Don Reble, Nov 17 2005

A100422 Numbers n such that 30*n+{1,7,11,13,17,23,29} are all prime.

Original entry on oeis.org

1, 53887, 114731, 123306, 139742, 210554, 471745, 480859, 619039, 630862, 858929, 1075873, 1306614, 1714945, 1913514, 2767458, 3014285, 3454137, 3518243, 3699151, 3864512, 3874291, 4274376, 4862362, 4878329, 4937822

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004

Values are 1 mod 7.

In each case, the 7 primes are necessarily consecutive. See the comment in A100418. - Peter Munn, Sep 06 2023

Cf. A005776, A007775, A076205, A100418-A100423.

[ n: n in [0..5000000] | forall{ q: q in [1, 7, 11, 13, 17, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 23 2011

a:= proc(n) option remember;
      local m;
      if n=1 then 1
      else for m from 30*(a(n-1)+7) by 210
           while not (isprime (m+1) and isprime (m+7) and
                 isprime (m+11) and isprime (m+13) and
                 isprime (m+17) and isprime (m+23) and
                 isprime (m+29))
           do od; m/30
        fi
    end:
seq (a(n), n=1..10);

Select[Range[5000000],And@@PrimeQ/@(30(#)+{1,7,11,13,17,23,29})&]  (* Harvey P. Dale, Feb 23 2011 *)

Edited by Don Reble, Nov 17 2005

A100421 Numbers n such that 30*n+{1,7,11,13,19,23,29} are all prime.

Original entry on oeis.org

2, 79, 391701, 505017, 740413, 787187, 933025, 1169863, 1333719, 1406792, 2212261, 2719950, 2962738, 3125992, 3284955, 3384586, 3727271, 3821295, 3861881, 4320864, 4439878, 4764356, 5014865, 5480190, 5879274, 6124442

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004

Values are 2 mod 7.

In each case, the 7 primes are necessarily consecutive. See the comment in A100418. - Peter Munn, Sep 06 2023

Cf. A005776, A007775, A076205, A100418-A100420, A100422, A100423.

[ n: n in [2..70000000 by 7] | forall{ q: q in [1, 7, 11, 13, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[7*10^6],AllTrue[30#+{1,7,11,13,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 16 2016 *)

Edited by Don Reble, Nov 17 2005

A385124 Numbers k such that there are exactly 7 primes between 30k and 30k+30.

Original entry on oeis.org

1, 2, 49, 62, 79, 89, 188, 6627, 9491, 18674, 22621, 31982, 34083, 38226, 38520, 41545, 48713, 53887, 89459, 103205, 114731, 123306, 139742, 140609, 149125, 168237, 175125, 210554, 223949, 229269, 237794, 240007, 267356, 288467, 321451, 364921, 368248, 373370, 391701

Offset: 1

Jianglin Luo, Jun 18 2025

nonn

The count of primes in 30*k..30*k+30 is less than 8 for k >= 1.

It appears that this sequence has infinitely many terms.

			1 is a term since there are 7 primes in 30..60: 31, 37, 41, 43, 47, 53, 59.
2 is a term since there are 7 primes in 60..90: 61, 67, 71, 73, 79, 83, 89.
3 is not a term since there are only 6 primes in 90..120: 97, 101, 103, 107, 109, 113.
49 is a term since there are 7 primes in 30*49..30*50: 1471, 1481, 1483, 1487, 1489, 1493, 1499.

Jianglin Luo, Table of n, a(n) for n = 1..3500

Union of A100418, A100419, A100420, A100421, A100422 and A100423.

Cf. A000720, A098592.

ArrayPlot[Table[Boole@PrimeQ[i*30+j],{i,0,399},{j,30}],Mesh->True]
index=1;Do[If[Length@(*PrimeRange=*) Select[Range[30*k+1,30*k+30,2],PrimeQ]==7,Print[index++," ",k]],{k,1,10^9}]

[n|n<-[1..10^6],#primes([30*n,30*n+30])==7]

{k | A098592(k) = pi(30*k+30) - pi(30*k) = 7}. - Michael S. Branicky, Jun 24 2025

cp's OEIS Frontend

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

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A100418 Numbers k such that 30*k + {1,11,13,17,19,23,29} are all prime.

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A076205 Numbers n such that 30*n+{1,7,11,13,17,19,23,29} are all composite.

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A100419 Numbers k such that 30*k+{1,7,13,17,19,23,29} are all prime.

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A100420 Numbers n such that 30*n+{1,7,11,17,19,23,29} are all prime.

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A100422 Numbers n such that 30*n+{1,7,11,13,17,23,29} are all prime.

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A100421 Numbers n such that 30*n+{1,7,11,13,19,23,29} are all prime.

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A385124 Numbers k such that there are exactly 7 primes between 30k and 30k+30.