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A100418
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Numbers k such that 30*k + {1,11,13,17,19,23,29} are all prime.
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10
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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
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OFFSET
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1,1
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COMMENTS
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Values are 0 mod 7.
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)
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004
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EXTENSIONS
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STATUS
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approved
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