A076205 -id:A076205 - cp's OEIS frontend

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

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

A100423 Numbers n such that 30*n+{1,7,11,13,17,19,29} are all prime.

Original entry on oeis.org

62, 188, 9491, 31982, 38226, 38520, 89459, 168237, 175125, 368248, 471078, 634892, 704416, 803102, 994748, 1436315, 1488857, 1605484, 1842553, 1945824, 2282958, 2465266, 2620715, 2627029, 2705037, 4282305, 5569899, 5914824

Offset: 1

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

Values are 6 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-A100422.

[ n: n in [0..6000000] | forall{ q: q in [1, 7, 11, 13, 17, 19, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[6*10^6],AllTrue[30#+{1,7,11,13,17,19,29},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 21 2021 *)

Edited by Don Reble, Nov 17 2005

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

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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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A100423 Numbers n such that 30*n+{1,7,11,13,17,19,29} are all prime.

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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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Magma

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Extensions