A022008 Initial member of prime sextuples (p, p+4, p+6, p+10, p+12, p+16).

7, 97, 16057, 19417, 43777, 1091257, 1615837, 1954357, 2822707, 2839927, 3243337, 3400207, 6005887, 6503587, 7187767, 7641367, 8061997, 8741137, 10526557, 11086837, 11664547, 14520547, 14812867, 14834707, 14856757, 16025827, 16094707, 18916477, 19197247

Offset

1,1

Comments

Without the initial 7, this gives primes at which difference pattern X42424Y (X and Y >= 8) occurs in A001223. - Labos Elemer

Subsequence of A022007. - Zak Seidov, Nov 01 2011

From Jean-Christophe Hervé, Sep 27 2014: (Start)

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).

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.

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)

All terms are congruent to 7 (mod 30). - Zak Seidov, May 07 2017

All terms but the first one are congruent to 97 (mod 210). - M. F. Hasler, Jan 18 2022

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov)

T. Forbes and Norman Luhn, Prime k-tuplets

R. J. Mathar, Table of Prime Gap Constellations

Example

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).

Maple

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

Mathematica

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
Transpose[Select[Partition[Prime[Range[10^6]], 6, 1], Differences[#]=={4, 2, 4, 2, 4}&]][[1]] (* Harvey P. Dale, Mar 15 2015 *)

Prog

(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
(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
(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
(Perl) use ntheory ":all"; say for sieve_prime_cluster(1, 1e8, 4, 6, 10, 12, 16); # Dana Jacobsen, Sep 30 2015
(GAP)
P:=Filtered([1, 3..2*10^7+1], IsPrime);;  I:=[4, 2, 4, 2, 4];;
P1:=List([1..Length(P)-1], i->P[i+1]-P[i]);;
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

Crossrefs

Cf. A022007.

Cf. A001223, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A047078.

Cf. A350826 (number of n-digit terms).

Sequence in context: A333246 A335922 A157035 * A200504 A267641 A267669

Adjacent sequences: A022005 A022006 A022007 * A022009 A022010 A022011

Keyword

nonn

Author

Warut Roonguthai

Status

approved