A143203 Numbers having exactly two distinct prime factors p, q with q = p+4.

21, 63, 77, 147, 189, 221, 437, 441, 539, 567, 847, 1029, 1323, 1517, 1701, 2021, 2873, 3087, 3757, 3773, 3969, 4757, 5103, 5929, 6557, 7203, 8303, 9261, 9317, 9797, 10051, 11021, 11907, 12317, 15309, 16637, 21609

Offset: 1

Author

Reinhard Zumkeller, Aug 12 2008

Keywords

• nonn

Comments

Subsequence of A007774.

A033850 is a subsequence.

Subsequence of A195106. - Reinhard Zumkeller, Sep 13 2011

Examples

			a(1) = 21 = 3 * 7 = A023200(1) * A046132(1).
a(2) = 63 = 3^2 * 7 = A023200(1)^2 * A046132(1).
a(3) = 77 = 7 * 11 = A023200(2) * A046132(2).
a(4) = 147 = 3 * 7^2 = A023200(1) * A046132(1)^2.
a(5) = 189 = 3*3 * 7 = A023200(1)^3 * A046132(1).
a(6) = 221 = 13 * 17 = A023200(3) * A046132(3).
a(7) = 437 = 19 * 23 = A023200(4) * A046132(4).
a(8) = 441 = 3^2 * 7^2 = A023200(1)^2 * A046132(1)^2.
a(9) = 539 = 7^2 * 11 = A023200(2)^2 * A046132(2).
a(10) = 567 = 3^4 * 7 = A023200(1)^4 * A046132(1).
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..250
• Eric Weisstein's World of Mathematics, Cousin Primes.
• Index entries for primes, gaps between.

Crossrefs

Cf. A001221, A006530, A007774, A020639, A023200, A027748, A033850, A046132, A143201, A195106.

Programs

Haskell

a143203 n = a143203_list !! (n-1)
a143203_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 4 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011

Mathematica

dpf2Q[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==2&&fi[[2]]-fi[[1]]==4]; Select[Range[22000],dpf2Q] (* Harvey P. Dale, Mar 18 2023 *)

Formula

A143201(a(n)) = 5.

A020639(a(n)) in A023200 and A006530(a(n)) in A046132.

A001221(a(n)) = 2.

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023200(n)+1)^2-4) = 0.109882433872... . - Amiram Eldar, Oct 26 2024