A022009 Initial members of prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20).

11, 165701, 1068701, 11900501, 15760091, 18504371, 21036131, 25658441, 39431921, 45002591, 67816361, 86818211, 93625991, 124716071, 136261241, 140117051, 154635191, 162189101, 182403491, 186484211, 187029371, 190514321, 198453371

Offset: 1

Author

Warut Roonguthai

Keywords

• nonn

Comments

All terms are congruent to 11 (modulo 210). - Matt C. Anderson, May 26 2015

Also the terms k of A276848 for which k == 1 (mod 10), see the comment in A276848 and A276826. All terms are obviously also congruent to 11 (modulo 30). - Vladimir Shevelev, Sep 21 2016

See A343637 for the least prime septuplets > 10^n, n >= 0. - M. F. Hasler, Aug 04 2021

Links

• Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Matt C. Anderson)
• Matt C. Anderson, table of prime k-tuplets.
• Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
• Norman Luhn, 1 million terms, zipped archive.
• Vladimir Shevelev and Peter J. C. Moses, Constellations of primes generated by twin primes, arXiv:1610.03385 [math.NT], 2016.
• Eric Weisstein's World of Mathematics, Prime Constellation.

Crossrefs

Cf. A022010 (septuplets of the second type), A182387, A276826, A276848, A343637 (septuplet following 10^n).

Programs

Magma

[p: p in PrimesUpTo(2*10^8) | forall{p+r: r in [2,6,8,12,18,20] | IsPrime(p+r)}]; // Vincenzo Librandi, Oct 01 2015

Mathematica

Transpose[Select[Partition[Prime[Range[10400000]],7,1],Differences[#] == {2,4,2,4,6,2}&]][[1]] (* Harvey P. Dale, Jul 13 2014 *)
Select[Prime[Range[2 10^8]], Union[PrimeQ[# + {2, 6, 8, 12, 18, 20}]] == {True} &] (* Vincenzo Librandi, Oct 01 2015 *)

PARI

nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n), n+1, n)
   is(n)=if(n%30!=11 || !isprime(n) || !isprime(n+2), return(0)); my(p=n, q=n+2, k=2, f); while(p!=q && q-p<7, f=if(isprime(k++), nextprime, nextcomposite); p=f(p+1); q=f(q+1)); p==q \\ Charles R Greathouse IV, Sep 30 2016

PARI

select( {is_A022009(n)=n%210==11&&!foreach([20,18,12,8,6,2,0],d,isprime(n+d)||return)}, [11+k*210|k<-[0..10^5]]) \\ M. F. Hasler, Aug 04 2021

Perl

use ntheory ":all"; say for sieve_prime_cluster(1,1e9, 2,6,8,12,18,20); # Dana Jacobsen, Sep 30 2015

Formula

a(n) = 210*A182387(n) + 11. - Hugo Pfoertner, Nov 18 2022