cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

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

Keywords

Comments

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)

Links

Crossrefs

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

Programs

  • Magma
    [ 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
  • Mathematica
    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 *)
  • PARI
    {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)}

Extensions

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

Views

Author

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

Keywords

Comments

Values are 4 mod 7.
In each case, the 7 primes are necessarily consecutive. See the comment in A100418. - Peter Munn, Sep 06 2023

Crossrefs

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

Programs

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

Extensions

Edited by Don Reble, Nov 17 2005

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

Views

Author

Jianglin Luo, Jun 18 2025

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Union of A100418, A100419, A100420, A100421, A100422 and A100423.
Cf. A000720, A098592.

Programs

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

Formula

{k | A098592(k) = pi(30*k+30) - pi(30*k) = 7}. - Michael S. Branicky, Jun 24 2025
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.