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

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, 48, 50, 52, 54, 56, 57, 58, 60, 64, 66, 68, 70, 72, 75, 77, 78, 81, 82, 84, 85, 88, 90, 91, 92, 93, 96, 98, 100, 102, 105, 106, 108, 110, 112, 114, 115, 117

Offset: 1

Author

N. J. A. Sloane, May 13 2002

Keywords

• nonn

Comments

Erdős conjectured that this sequence has asymptotic density 1/2.

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

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

References

• H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Régis De La Bretèche, Carl Pomerance and Gérald Tenenbaum, Products of ratios of consecutive integers, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 131-138, alternative link.
• Paul Erdős and Carl Pomerance, On the largest prime factors of n and n + 1, Aequationes Math., Vol. 17, No. 1 (1978), pp. 311-321, alternative link.
• Xiaodong Lü and Zhiwei Wang, On the largest prime factors of consecutive integers, 2018.
• Zhiwei Wang, On the largest prime factors of consecutive integers in short intervals, Proceedings of the American Mathematical Society, Vol. 145, No. 8 (2017), pp. 3211-3220.
• Zhiwei Wang, Sur les plus grands facteurs premiers d'entiers consécutifs, Mathematika, Vol. 64, No. 2 (2018), pp. 343-379, preprint, arXiv:1706.02980 [math.NT], 2017.

Crossrefs

Cf. A006530, A070087.

Programs

Mathematica

f[n_] := FactorInteger[n][[ -1, 1]]; Select[ Range[125], f[ # ] < f[ # + 1] &]

PARI

gpf(n)=if(n<3,n,my(f=factor(n)[,1]); f[#f])
is(n)=gpf(n) < gpf(n+1) \\ Charles R Greathouse IV, Oct 27 2015