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A144105 Primes at the upper end of the gaps mentioned in A144104. 2
3, 5, 11, 17, 29, 37, 53, 59, 127, 149, 211, 223, 307, 331, 541, 1361, 1693, 1973, 2203, 2503, 2999, 3299, 4327, 4861, 5623, 5779, 5981, 6521, 6947, 7283, 8501, 9587, 10007, 10831, 11777, 12197, 12889, 15727, 16183, 19661, 31469, 34123, 35671, 35729 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Firoozbakht conjecture: (prime(n+1))^(1/(n+1)) < prime(n)^(1/n), or

prime(n+1) < prime(n)^(1+1/n), which can be rewritten as: [log(prime(n+1))/log(prime(n))]^n < (1+1/n)^n. This suggests a weaker conjecture: [log(prime(n+1))/log(prime(n))]^n < e. - Daniel Forgues, Apr 28 2014

LINKS

T. D. Noe, Table of n, a(n) for n=1..176

A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015

Nilotpal Kanti Sinha, On a new property of primes that leads to a generalization of Cramer's conjecture, arXiv:1010.1399 [math.NT]

Wikipedia, Firoozbakht’s conjecture

EXAMPLE

Examples for (log(prime(n+1))/log(prime(n)))^n < (1+1/n)^n < e:

(log(3)/log(2))^1 = 1.58... < (1+1/1)^1 = 2 < e;

(log(1361)/log(1327))^217 = 2.14... < (1+1/217)^217 = 2.712... < e;

(log(8501)/log(8467))^1059 = 1.59... < (1+1/1059)^1059 = 2.716... < e;

(log(35729)/log(35677))^3795 = 1.69... < (1+1/3795)^3795 = 2.717... < e.

- Daniel Forgues, Apr 28 2014

CROSSREFS

Sequence in context: A147350 A066692 A123533 * A141262 A069233 A063700

Adjacent sequences:  A144102 A144103 A144104 * A144106 A144107 A144108

KEYWORD

nonn

AUTHOR

T. D. Noe, Sep 11 2008

STATUS

approved

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Last modified December 7 05:39 EST 2016. Contains 278841 sequences.