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A070089
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P(n) < P(n+1) where P(n) (A006530) is the largest prime factor of n.
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14
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.
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LINKS
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MATHEMATICA
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f[n_] := FactorInteger[n][[ -1, 1]]; Select[ Range[125], f[ # ] < f[ # + 1] &]
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PROG
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(PARI) gpf(n)=if(n<3, n, my(f=factor(n)[, 1]); f[#f])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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