OFFSET
1,1
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.
(End)
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10309
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 *)
PROG
(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)}
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004
EXTENSIONS
Edited by Don Reble, Nov 17 2005
STATUS
approved