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

Author

T. D. Noe, Sep 11 2008

Keywords

• nonn

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

Examples

			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
		

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