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A023287
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Primes that remain prime through 3 iterations of function f(x) = 6x + 1.
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4
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61, 101, 1811, 3491, 4091, 5711, 5801, 6361, 7121, 10391, 10771, 11311, 13421, 15131, 17791, 18911, 19471, 20011, 24391, 25601, 25951, 30091, 35251, 41911, 45631, 47431, 55631, 58711, 62921, 67891, 70451, 70571, 72271, 74051, 74161, 75431, 80471, 86341
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OFFSET
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1,1
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COMMENTS
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Primes p such that s1=p, s2=6*s1+1, s3=6*s2+1 and s4=6*s3+1 are primes forming a special chain of four primes. A fifth term in such a chain cannot arise. See A085956, A086361, A086362.
Entries in chains are congruent to {1,7,3,9} mod 10.
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LINKS
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FORMULA
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{p, 6p+1, 36p+7, 216p+43} are all primes, where p is prime.
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EXAMPLE
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First chain is {61, 367, 2203, 13219};
319th chain is {1291391, 7748347, 46490083, 278940499}.
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MATHEMATICA
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k=0; m=6; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; If[PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], k=k+1; Print[{k, n, s, s1, s2, s3}]], {n, 1, 100000}] (* edited by _Zak Seidov_, Feb 08 2011 *)
thrQ[n_]:=AllTrue[Rest[NestList[6#+1&, n , 3]], PrimeQ]; Select[Prime[Range[9000]], thrQ] (* _Harvey P. Dale_, Mar 03 2024 *)
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PROG
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(Magma) [n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7) and IsPrime(216*n+43)] // _Vincenzo Librandi_, Aug 04 2010
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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_David W. Wilson_
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EXTENSIONS
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Additional comments from _Labos Elemer_, Jul 23 2003
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STATUS
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approved
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