A143205 Numbers having exactly two distinct prime factors p, q with q = p+6.

55, 91, 187, 247, 275, 391, 605, 637, 667, 1147, 1183, 1375, 1591, 1927, 2057, 2491, 3025, 3127, 3179, 3211, 4087, 4459, 4693, 4891, 5767, 6647, 6655, 6875, 7387, 8281, 8993, 9991, 10807, 11227, 12091, 15125, 15341, 15379, 17947, 19343, 22627, 23707

Offset: 1

Author

Reinhard Zumkeller, Jul 30 2008

Keywords

• nonn

Comments

Subsequence of A007774.

A111192 is a subsequence.

Subsequence of A195118. - Reinhard Zumkeller, Sep 13 2011

Examples

			a(1) = 55 = 5 * 11 = A023201(1) * A046117(1).
a(2) = 91 = 7 * 13 = A023201(2) * A046117(2).
a(3) = 187 = 11 * 17 = A023201(3) * A046117(3).
a(4) = 247 = 13 * 19 = A023201(4) * A046117(4).
a(5) = 275 = 5^2 * 11 = A023201(1)^2 * A046117(1).
a(6) = 391 = 17 * 23 = A023201(5) * A046117(5).
a(7) = 605 = 5 * 11^2 = A023201(1) * A046117(1)^2.
a(8) = 637 = 7^2 * 13 = A023201(2)^2 * A046117(2).
a(9) = 667 = 23 * 29 = A023201(6) * A046117(6).
a(10) = 1147 = 31 * 37 = A023201(7) * A046117(7).
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..250
• Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]
• Index entries for primes, gaps between.

Crossrefs

Cf. A001221, A006530, A007774, A020639, A023201, A027748, A046117, A111192, A143201, A195118.

Programs

Haskell

a143205 n = a143205_list !! (n-1)
a143205_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 6 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011

Mathematica

okQ[n_]:=Module[{fi=Transpose[FactorInteger[n]][[1]]},Length[fi]==2 && Last[fi]-First[fi]==6]; Select[Range[25000],okQ]  (* Harvey P. Dale, Apr 18 2011 *)

Formula

A143201(a(n)) = 7.

A020639(a(n)) in A023201 and A006530(a(n)) in A046117.

A001221(a(n)) = 2.

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023201(n)+2)^2-9) = 0.058842810164... . - Amiram Eldar, Oct 26 2024