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

%I A143202 #22 Feb 16 2025 08:33:08
%S A143202 15,35,45,75,135,143,175,225,245,323,375,405,675,875,899,1125,1215,
%T A143202 1225,1573,1715,1763,1859,1875,2025,3375,3599,3645,4375,5183,5491,
%U A143202 5625,6075,6125,6137,8575,9375,10125,10403,10935,11663,12005,16875,17303,18225
%N A143202 Numbers having exactly two distinct prime factors p, q with q = p+2.
%C A143202 Subsequence of A007774.
%C A143202 A037074 is a subsequence.
%H A143202 Reinhard Zumkeller, <a href="/A143202/b143202.txt">Table of n, a(n) for n = 1..250</a>
%H A143202 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>.
%H A143202 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>.
%F A143202 A143201(a(n)) = 3.
%F A143202 A020639(a(n)) in A001359 and A006530(a(n)) in A006512.
%F A143202 A001221(a(n)) = 2.
%F A143202 Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/(A001359(n)^2-1) = 0.1812568234997... . - _Amiram Eldar_, Oct 26 2024
%e A143202 a(1) = 15 = 3 * 5 = A001359(1) * A006512(1).
%e A143202 a(2) = 35 = 5 * 7 = A001359(2) * A006512(2).
%e A143202 a(3) = 45 = 3^2 * 5 = A001359(1)^2 * A006512(1).
%e A143202 a(4) = 75 = 3 * 5^2 = A001359(1) * A006512(1)^2.
%e A143202 a(5) = 135 = 3^3 * 5 = A001359(1)^3 * A006512(1).
%e A143202 a(6) = 143 = 11 * 13 = A001359(3) * A006512(3).
%e A143202 a(7) = 175 = 5^2 * 7 = A001359(2)^2 * A006512(2).
%e A143202 a(8) = 225 = 3^2 * 5^2 = A001359(1)^2 * A006512(1)^2.
%e A143202 a(9) = 245 = 5 * 7^2 = A001359(2) * A006512(2)^2.
%e A143202 a(10) = 323 = 17 * 19 = A001359(4) * A006512(4).
%e A143202 a(11) = 375 = 3 * 5^3 = A001359(1) * A006512(1)^3.
%e A143202 a(12) = 405 = 3^4 * 5 = A001359(1)^4 * A006512(1).
%t A143202 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 *)
%o A143202 (Haskell)
%o A143202 a143202 n = a143202_list !! (n-1)
%o A143202 a143202_list = filter (\x -> a006530 x - a020639 x == 2) [1,3..]
%o A143202 -- _Reinhard Zumkeller_, Sep 13 2011
%Y A143202 Cf. A001221, A001359, A006512, A006530, A007774, A020639, A037074, A143201.
%K A143202 nonn
%O A143202 1,1
%A A143202 _Reinhard Zumkeller_, Aug 12 2008