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

Author

Eric W. Weisstein

Keywords

• nonn
• nice
• hard

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.

Examples

			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}.
		

Links

• 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.

Crossrefs

Cf. A007530.

Programs

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 *)

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