A144105 Primes at the upper end of the gaps mentioned in A144104.

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

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], 2010.

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 A329946

Adjacent sequences: A144102 A144103 A144104 * A144106 A144107 A144108

Keyword

nonn

Author

T. D. Noe, Sep 11 2008

Status

approved