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

Author

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

Keywords

• easy
• nonn

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

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

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