cp's OEIS Frontend

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

%I A143203 #23 Feb 16 2025 08:33:08
%S A143203 21,63,77,147,189,221,437,441,539,567,847,1029,1323,1517,1701,2021,
%T A143203 2873,3087,3757,3773,3969,4757,5103,5929,6557,7203,8303,9261,9317,
%U A143203 9797,10051,11021,11907,12317,15309,16637,21609
%N A143203 Numbers having exactly two distinct prime factors p, q with q = p+4.
%C A143203 Subsequence of A007774.
%C A143203 A033850 is a subsequence.
%C A143203 Subsequence of A195106. - _Reinhard Zumkeller_, Sep 13 2011
%H A143203 Reinhard Zumkeller, <a href="/A143203/b143203.txt">Table of n, a(n) for n = 1..250</a>
%H A143203 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CousinPrimes.html">Cousin Primes</a>.
%H A143203 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>.
%F A143203 A143201(a(n)) = 5.
%F A143203 A020639(a(n)) in A023200 and A006530(a(n)) in A046132.
%F A143203 A001221(a(n)) = 2.
%F A143203 Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023200(n)+1)^2-4) = 0.109882433872... . - _Amiram Eldar_, Oct 26 2024
%e A143203 a(1) = 21 = 3 * 7 = A023200(1) * A046132(1).
%e A143203 a(2) = 63 = 3^2 * 7 = A023200(1)^2 * A046132(1).
%e A143203 a(3) = 77 = 7 * 11 = A023200(2) * A046132(2).
%e A143203 a(4) = 147 = 3 * 7^2 = A023200(1) * A046132(1)^2.
%e A143203 a(5) = 189 = 3*3 * 7 = A023200(1)^3 * A046132(1).
%e A143203 a(6) = 221 = 13 * 17 = A023200(3) * A046132(3).
%e A143203 a(7) = 437 = 19 * 23 = A023200(4) * A046132(4).
%e A143203 a(8) = 441 = 3^2 * 7^2 = A023200(1)^2 * A046132(1)^2.
%e A143203 a(9) = 539 = 7^2 * 11 = A023200(2)^2 * A046132(2).
%e A143203 a(10) = 567 = 3^4 * 7 = A023200(1)^4 * A046132(1).
%t A143203 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 *)
%o A143203 (Haskell)
%o A143203 a143203 n = a143203_list !! (n-1)
%o A143203 a143203_list = filter f [1,3..] where
%o A143203    f x = length pfs == 2 && last pfs - head pfs == 4 where
%o A143203        pfs = a027748_row x
%o A143203 -- _Reinhard Zumkeller_, Sep 13 2011
%Y A143203 Cf. A001221, A006530, A007774, A020639, A023200, A027748, A033850, A046132, A143201, A195106.
%K A143203 nonn
%O A143203 1,1
%A A143203 _Reinhard Zumkeller_, Aug 12 2008