A022010 Initial members of prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).

5639, 88799, 284729, 626609, 855719, 1146779, 6560999, 7540439, 8573429, 17843459, 19089599, 24001709, 42981929, 43534019, 69156539, 74266259, 79208399, 80427029, 84104549, 87988709, 124066079, 128469149, 144214319, 157131419, 208729049, 218033729

Offset

1,1

Comments

All terms are congruent to 179 (modulo 210). - Matt C. Anderson, May 26 2015

Links

Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Matt C. Anderson)

Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.

Norman Luhn, 1 million terms (zipped archive).

Hugo Pfoertner, Illustration of n/Integral_{x=2,a(n)} 1/log(x)^7 dx approaching Hardy-Littlewood bound. (2020).

Formula

a(n) = 210*A357889(n) + 179. - Hugo Pfoertner, Nov 18 2022

Example

a(100) = 2526962939, a(1000) = 80752495919, a(10000) = 2010407120789, a(100000) = 42609827234069, a(1000000) = 822249634821059. See illustration for asymptotic behavior. - Hugo Pfoertner, Jun 15 2020

Mathematica

Select[Prime[Range[2 10^8]], Union[PrimeQ[# + {2, 8, 12, 14, 18, 20}]] == {True} &] (* Vincenzo Librandi, Oct 01 2015 *)
Select[Partition[Prime[Range[12021000]], 7, 1], Differences[#]=={2, 6, 4, 2, 4, 2}&][[All, 1]] (* or *) Select[Range[179, 219*10^6, 210], AllTrue[ #+{0, 2, 8, 12, 14, 18, 20}, PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 04 2019 *)

Prog

(Perl) use ntheory ":all"; say for sieve_prime_cluster(1, 1e9, 2, 8, 12, 14, 18, 20); # Dana Jacobsen, Sep 30 2015
(Magma) [p: p in PrimesUpTo(3*10^8) | forall{p+r: r in [2, 8, 12, 14, 18, 20] | IsPrime(p+r)}]; // Vincenzo Librandi, Oct 01 2015
(PARI) forprime(p=2, 10^30, if (isprime(p+2) && isprime(p+8) && isprime(p+12) && isprime(p+14) && isprime(p+18) && isprime(p+20), print1(p", "))) \\ Altug Alkan, Oct 01 2015. [This can be made 2x faster by inserting "p%210==179 &&" before or after "if(". - M. F. Hasler, Aug 04 2021]

Crossrefs

Cf. A022009 (prime septuplets of the first type), A332493.

Cf. A257124 (union of this and A022009), A343637 (septuplet following 10^n).

Cf. A357889.

Sequence in context: A045128 A229591 A161193 * A201252 A247402 A339961

Adjacent sequences: A022007 A022008 A022009 * A022011 A022012 A022013

Keyword

nonn

Author

Warut Roonguthai

Extensions

More terms from a Maple program by Matt C. Anderson, Dec 05 2013

Status

approved