A143202 Numbers having exactly two distinct prime factors p, q with q = p+2.

15, 35, 45, 75, 135, 143, 175, 225, 245, 323, 375, 405, 675, 875, 899, 1125, 1215, 1225, 1573, 1715, 1763, 1859, 1875, 2025, 3375, 3599, 3645, 4375, 5183, 5491, 5625, 6075, 6125, 6137, 8575, 9375, 10125, 10403, 10935, 11663, 12005, 16875, 17303, 18225

Offset: 1

Author

Reinhard Zumkeller, Aug 12 2008

Keywords

• nonn

Comments

Subsequence of A007774.

A037074 is a subsequence.

Examples

			a(1) = 15 = 3 * 5 = A001359(1) * A006512(1).
a(2) = 35 = 5 * 7 = A001359(2) * A006512(2).
a(3) = 45 = 3^2 * 5 = A001359(1)^2 * A006512(1).
a(4) = 75 = 3 * 5^2 = A001359(1) * A006512(1)^2.
a(5) = 135 = 3^3 * 5 = A001359(1)^3 * A006512(1).
a(6) = 143 = 11 * 13 = A001359(3) * A006512(3).
a(7) = 175 = 5^2 * 7 = A001359(2)^2 * A006512(2).
a(8) = 225 = 3^2 * 5^2 = A001359(1)^2 * A006512(1)^2.
a(9) = 245 = 5 * 7^2 = A001359(2) * A006512(2)^2.
a(10) = 323 = 17 * 19 = A001359(4) * A006512(4).
a(11) = 375 = 3 * 5^3 = A001359(1) * A006512(1)^3.
a(12) = 405 = 3^4 * 5 = A001359(1)^4 * A006512(1).
		

Links

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

Crossrefs

Cf. A001221, A001359, A006512, A006530, A007774, A020639, A037074, A143201.

Programs

Haskell

a143202 n = a143202_list !! (n-1)
a143202_list = filter (\x -> a006530 x - a020639 x == 2) [1,3..]
-- Reinhard Zumkeller, Sep 13 2011

Mathematica

tdpfQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==2&&fi[[2]]-fi[[1]]==2]; Select[Range[20000],tdpfQ] (* Harvey P. Dale, Mar 04 2023 *)

Formula

A143201(a(n)) = 3.

A020639(a(n)) in A001359 and A006530(a(n)) in A006512.

A001221(a(n)) = 2.

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/(A001359(n)^2-1) = 0.1812568234997... . - Amiram Eldar, Oct 26 2024