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%I #34 Nov 02 2020 09:55:39
%S 13,14,34,37,38,43,61,62,73,79,86,94,103,118,122,123,142,151,152,157,
%T 158,163,173,185,193,194,202,206,214,218,223,229,241,254,257,258,271,
%U 277,278,283,284,295,298,302,313,317,318,321,322,326,331,334,341,373
%N Numbers k such that gpf(k) > gpf(k+1) > gpf(k+2) where gpf(k) denotes the largest prime factor of k.
%C Erdős conjectured that this sequence is infinite.
%C Balog (2001) proved that this sequence is infinite. - _Amiram Eldar_, Aug 02 2020
%H T. D. Noe, <a href="/A071870/b071870.txt">Table of n, a(n) for n = 1..1000</a>
%H Antal Balog, <a href="https://doi.org/10.1556/sscmath.38.2001.1-4.3">On triplets with descending largest prime factors</a>, Studia Scientiarum Mathematicarum Hungarica, Vol. 38, No. 1-4 (2001), pp. 45-50.
%H P. Erdős and C. Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/PDF/paper17.pdf">On the largest prime factors of n and n+1</a>, Aequationes Math. 17 (1978), pp. 311-321. [<a href="http://www.renyi.hu/~p_erdos/1978-29.pdf">alternate link</a>]
%H Sungjin Kim, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kim/kim22.html">Two Remarks on the Largest Prime Factors of n and n+1</a>, J. of Integer Sequences, 23 (2020), #20.10.1.
%e 13 is a term since gpf(13) = 13, gpf(14) = 7, gpf(15) = 5, and 13 > 7 > 5.
%t Select[ Range[400], FactorInteger[#][[-1, 1]] > FactorInteger[# + 1][[-1, 1]] > FactorInteger[# + 2][[-1, 1]] &] (* _Jean-François Alcover_, Jun 17 2013 *)
%o (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 (PARI) gpf(n) = vecmax(factor(n)[,1]);
%o isok(k) = (gpf(k) > gpf(k+1)) && (gpf(k+1) > gpf(k+2)); \\ _Michel Marcus_, Nov 02 2020
%o (Python)
%o from sympy import factorint
%o A071870_list, p, q, r = [], 1, 2, 3
%o for n in range(2,10**4):
%o p, q, r = q, r, max(factorint(n+2))
%o if p > q > r:
%o A071870_list.append(n) # _Chai Wah Wu_, Jul 24 2017
%Y Cf. A006530, A070087, A071869, A082417, A082418, A082419, A082420, A082421, A082422.
%K nonn
%O 1,1
%A _Benoit Cloitre_, Jun 09 2002
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