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A070089 P(n) < P(n+1) where P(n) (A006530) is the largest prime factor of n.

%I A070089 #17 Aug 02 2020 04:09:02
%S A070089 1,2,4,6,8,9,10,12,16,18,20,21,22,24,25,27,28,30,32,33,36,40,42,45,46,
%T A070089 48,50,52,54,56,57,58,60,64,66,68,70,72,75,77,78,81,82,84,85,88,90,91,
%U A070089 92,93,96,98,100,102,105,106,108,110,112,114,115,117
%N A070089 P(n) < P(n+1) where P(n) (A006530) is the largest prime factor of n.
%C A070089 Erdős conjectured that this sequence has asymptotic density 1/2.
%C A070089 There are 500149 terms in this sequence up to 10^6, 4999951 up to 10^7, 49997566 up to 10^8, and 499992458 up to 10^9. With a binomial model with p = 1/2, these would be +0.3, -0.5, -0.0, and -0.5 standard deviations from their respective means. In other words, Erdős's conjecture seems solid. - _Charles R Greathouse IV_, Oct 27 2015
%C A070089 Erdős and Pomerance (1978) proved that the lower density of this sequence is at least 0.0099. This value was improved to 0.05544 (De La Bretèche et al., 2005), 0.1063 (Wang, 2017), 0.1356 (Wang, 2018), and 0.2017 (Lü and Wang, 2018). - _Amiram Eldar_, Aug 02 2020
%D A070089 H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.
%H A070089 T. D. Noe, <a href="/A070089/b070089.txt">Table of n, a(n) for n = 1..1000</a>
%H A070089 Régis De La Bretèche, Carl Pomerance and Gérald Tenenbaum, <a href="https://doi.org/10.1007/s11139-005-0831-7">Products of ratios of consecutive integers</a>, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 131-138, <a href="https://math.dartmouth.edu/~carlp/PDF/ABtemp.pdf">alternative link</a>.
%H A070089 Paul Erdős and Carl Pomerance, <a href="https://doi.org/10.1007/BF01818569">On the largest prime factors of n and n + 1</a>, Aequationes Math., Vol. 17, No. 1 (1978), pp. 311-321, <a href="http://www.math.dartmouth.edu/~carlp/PDF/paper17.pdf">alternative link</a>.
%H A070089 Xiaodong Lü and Zhiwei Wang, <a href="https://hal.archives-ouvertes.fr/hal-01797939/">On the largest prime factors of consecutive integers</a>, 2018.
%H A070089 Zhiwei Wang, <a href="https://doi.org/10.1090/proc/13459">On the largest prime factors of consecutive integers in short intervals</a>, Proceedings of the American Mathematical Society, Vol. 145, No. 8 (2017), pp. 3211-3220.
%H A070089 Zhiwei Wang, <a href="https://doi.org/10.1112/S0025579317000547">Sur les plus grands facteurs premiers d'entiers consécutifs</a>, Mathematika, Vol. 64, No. 2 (2018), pp. 343-379, <a href="https://arxiv.org/abs/1706.02980">preprint</a>, arXiv:1706.02980 [math.NT], 2017.
%t A070089 f[n_] := FactorInteger[n][[ -1, 1]]; Select[ Range[125], f[ # ] < f[ # + 1] &]
%o A070089 (PARI) gpf(n)=if(n<3,n,my(f=factor(n)[,1]); f[#f])
%o A070089 is(n)=gpf(n) < gpf(n+1) \\ _Charles R Greathouse IV_, Oct 27 2015
%Y A070089 Cf. A006530, A070087.
%K A070089 nonn
%O A070089 1,2
%A A070089 _N. J. A. Sloane_, May 13 2002