A050258 Number of "prime quadruplets" with largest member < 10^n.

0, 2, 5, 12, 38, 166, 899, 4768, 28388, 180529, 1209318, 8398278, 60070590, 441296836, 3314576487, 25379433651, 197622677481

Offset

1,2

Comments

A "prime quadruplet" is a set of four primes {p, p+2, p+6, p+8}.

a(1) = 0 rather than 1 because the quadruple {2,3,5,7} does not have the official form.

Links

Table of n, a(n) for n=1..17.

Thomas R. Nicely, Enumeration of the prime quadruplets to 1e16.

Thomas R. Nicely, Enumeration to 1.6e15 of the prime quadruplets.

Jonathan P. Sorenson, Jonathan Webster, Two Algorithms to Find Primes in Patterns, arXiv:1807.08777 [math.NT], 2018.

Eric Weisstein's World of Mathematics, Prime Quadruplet.

Index entries for sequences related to numbers of primes in various ranges.

Example

a(2) = 2 because there are two prime quadruplets with largest member less than 10^2, namely {5, 7, 11, 13} and {11, 13, 17, 19}.
a(3) = 5 because, in addition to the prime quadruplets mentioned above, below 10^3 we also have {101, 103, 107, 109}, {191, 193, 197, 199} and {821, 823, 827, 829}.

Mathematica

c = 1; Do[ Do[ If[ PrimeQ[ n ] && PrimeQ[ n + 2 ] && PrimeQ[ n + 6 ] && PrimeQ[ n + 8 ], c++ ], {n, 10^n + 1, 10^(n + 1), 10} ]; Print[ c ], {n, 1, 15} ] (* Weisstein *)
(* First run program for A090258 *) Table[Length[Select[A090258, # < 10^n &]], {n, 5}] (* Alonso del Arte, Aug 12 2012 *)

Crossrefs

Cf. A007530.

Sequence in context: A130221 A036782 A050237 * A051436 A054581 A203151

Adjacent sequences: A050255 A050256 A050257 * A050259 A050260 A050261

Keyword

nonn,nice,hard

Author

Eric W. Weisstein

Extensions

a(16) (from Nicely link) added by Donovan Johnson, Jan 11 2011

a(17) added by Jonathan Webster, Jun 26 2018

a(1) changed to 0 at the suggestion of Harvey P. Dale. - N. J. A. Sloane, Sep 25 2019

Status

approved