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

%I A100418 #23 Sep 06 2023 08:09:13
%S A100418 49,34083,41545,48713,140609,524027,616812,855281,1314397,1324750,
%T A100418 1636152,2281293,2927134,3401412,3605413,4989341,5212221,5284979,
%U A100418 5406303,5645269,6141254,6342728,7231434,7347697,7637329,8027068,8161657,8372756,8392776,8567216,8986096,9145563
%N A100418 Numbers k such that 30*k + {1,11,13,17,19,23,29} are all prime.
%C A100418 Values are 0 mod 7.
%C A100418 From _Peter Munn_, Sep 06 2023: (Start)
%C A100418 In each case, the 7 primes are necessarily consecutive.
%C A100418 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.
%C A100418 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.
%C A100418 The sequences for the other appropriate subsets of 7 coprime values are A100419-A100423.
%C A100418 (End)
%H A100418 David A. Corneth, <a href="/A100418/b100418.txt">Table of n, a(n) for n = 1..10309</a>
%t A100418 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 *)
%o A100418 (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)}
%o A100418 (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
%Y A100418 Cf. A005776, A007775, A056956, A065706, A076205, A100419-A100423, A364678.
%K A100418 easy,nonn
%O A100418 1,1
%A A100418 Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004
%E A100418 Edited by _Don Reble_, Nov 17 2005