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%I #89 Feb 21 2022 02:17:40
%S 7,97,16057,19417,43777,1091257,1615837,1954357,2822707,2839927,
%T 3243337,3400207,6005887,6503587,7187767,7641367,8061997,8741137,
%U 10526557,11086837,11664547,14520547,14812867,14834707,14856757,16025827,16094707,18916477,19197247
%N Initial member of prime sextuples (p, p+4, p+6, p+10, p+12, p+16).
%C Without the initial 7, this gives primes at which difference pattern X42424Y (X and Y >= 8) occurs in A001223. - _Labos Elemer_
%C Subsequence of A022007. - _Zak Seidov_, Nov 01 2011
%C From _Jean-Christophe Hervé_, Sep 27 2014: (Start)
%C The primes in a sextuple a(n), a(n)+4, a(n)+6, a(n)+10, a(n)+12, a(n)+16 are consecutive since a(n)+2, a(n)+8 and a(n)+14 cannot be prime (multiple of 3).
%C The prime sextuples starting at a(n) give the highest concentration of primes that can occur on an interval of 17 integers (apart intervals starting at p < 7). It is conjectured that there are infinitely many such sextuples.
%C For n > 1, the 3 odd integers preceding and the 3 odd integers following the sextuple are not prime: a(n)-2 == a(n)+18 == 0 (mod 5), a(n)-4 == a(n)+20 == 0 (mod 3), a(n)-6 == a(n)+22 == 0 (mod 7) and thus a(n) == 97 (mod 210 = 2*3*5*7). (End)
%C All terms are congruent to 7 (mod 30). - _Zak Seidov_, May 07 2017
%C All terms but the first one are congruent to 97 (mod 210). - _M. F. Hasler_, Jan 18 2022
%H David A. Corneth, <a href="/A022008/b022008.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Zak Seidov)
%H T. Forbes and Norman Luhn, <a href="http://www.pzktupel.de/ktuplets">Prime k-tuplets</a>
%H R. J. Mathar, <a href="/A022004/a022004_1.pdf">Table of Prime Gap Constellations</a>
%e n=2: 97, 101, 103, 107, 109, 113 are consecutive primes, while 91, 93, 95 and 115, 117 and 119 are not (cf. 4th comment about the border of composites).
%p for i from 1 to 2*10^5 do if [ithprime(i+1), ithprime(i+2), ithprime(i+3), ithprime(i+4), ithprime(i+5)] = [ithprime(i)+4,ithprime(i)+6,ithprime(i)+10,ithprime(i)+12,ithprime(i)+16] then print(ithprime(i)); fi; od; # _Muniru A Asiru_, Sep 03 2017
%t lst = {}; Do[p = Prime[n]; If[PrimeQ[p+4] && PrimeQ[p+6] && PrimeQ[p+10] && PrimeQ[p+12] && PrimeQ[p+16], AppendTo[lst, p]], {n, 1000000}]; lst
%t Transpose[Select[Partition[Prime[Range[10^6]],6,1],Differences[#]=={4,2,4,2,4}&]][[1]] (* _Harvey P. Dale_, Mar 15 2015 *)
%o (PARI) p=2;q=3;r=5;s=7;t=11;forprime(u=13,1e9,if(u-p==16 && p%3==1, print1(p", "));p=q;q=r;r=s;s=t;t=u) \\ _Charles R Greathouse IV_, Mar 29 2013
%o (PARI) {next_A022008(p, L=Vec(p+1,5), m=210, r=Mod(97,m))=for(i=1,oo, L[i%5+1]+16==(p=nextprime(p+1))&&break; p%m>111 && until(r==p=nextprime((p+8)\/210*210+97),); L[i%5+1]=p); p-16} \\ _M. F. Hasler_, Jan 18 2022
%o (Magma) [p: p in PrimesUpTo(2*10^7) | IsPrime(p+4) and IsPrime(p+6) and IsPrime(p+10)and IsPrime(p+12) and IsPrime(p+16)]; // _Vincenzo Librandi_, Aug 23 2015
%o (Perl) use ntheory ":all"; say for sieve_prime_cluster(1,1e8, 4,6,10,12,16); # _Dana Jacobsen_, Sep 30 2015
%o (GAP)
%o P:=Filtered([1,3..2*10^7+1],IsPrime);; I:=[4,2,4,2,4];;
%o P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
%o A022008:=List(Positions(List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4]]),I),j->P[j]); # _Muniru A Asiru_, Sep 03 2017
%Y Cf. A022007.
%Y Cf. A001223, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A047078.
%Y Cf. A350826 (number of n-digit terms).
%K nonn
%O 1,1
%A _Warut Roonguthai_
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