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A071869 Numbers k such that gpf(k) < gpf(k+1) < gpf(k+2) where gpf(k) denotes the largest prime factor of k.

%I A071869 #29 Jun 05 2022 08:30:00
%S A071869 8,9,20,21,24,27,32,45,56,57,77,81,84,90,91,92,105,114,120,125,132,
%T A071869 135,140,144,147,165,168,169,170,171,175,176,177,189,200,204,212,216,
%U A071869 220,221,225,231,234,235,247,252,260,261,275,288,289,300,315,324,345,354
%N A071869 Numbers k such that gpf(k) < gpf(k+1) < gpf(k+2) where gpf(k) denotes the largest prime factor of k.
%C A071869 Erdős and Pomerance showed in 1978 that this sequence is infinite.
%H A071869 T. D. Noe, <a href="/A071869/b071869.txt">Table of n, a(n) for n = 1..1000</a>
%H A071869 Paul Erdős and Carl Pomerance, <a href="http://www.renyi.hu/~p_erdos/1978-29.pdf">On the largest prime factors of n and n+1</a>, Aequationes Math. 17 (1978), pp. 311-321.
%F A071869 a(n) = A079747(n+1) - 1. - _T. D. Noe_, Nov 26 2007
%t A071869 gpf[n_] := FactorInteger[n][[-1, 1]]; ind = Position[Differences[Array[gpf, 350, 2]], _?(# > 0 &)] // Flatten; ind[[Position[Differences[ind], 1] // Flatten]] + 1 (* _Amiram Eldar_, Jun 05 2022 *)
%o A071869 (PARI) for(n=2,500,if(sign(component(component(factor(n),1),omega(n))-component(component(factor(n+1),1),omega(n+1)))+sign(component(component(factor(n+1),1),omega(n+1))-component(component(factor(n+2),1),omega(n+2)))==-2,print1(n,",")))
%o A071869 (Python)
%o A071869 from sympy import factorint
%o A071869 A071869_list, p, q, r = [], 1, 2, 3
%o A071869 for n in range(2,10**4):
%o A071869     p, q, r = q, r, max(factorint(n+2))
%o A071869     if p < q < r:
%o A071869         A071869_list.append(n) # _Chai Wah Wu_, Jul 24 2017
%Y A071869 Cf. A006530, A070089, A071870, A079747, A082417-A082422.
%K A071869 nonn
%O A071869 1,1
%A A071869 _Benoit Cloitre_, Jun 09 2002