A144104 Primes p such that log(nextPrime(p))/log(p) is smaller for larger primes.

2, 3, 7, 13, 23, 31, 47, 53, 113, 139, 199, 211, 293, 317, 523, 1327, 1669, 1951, 2179, 2477, 2971, 3271, 4297, 4831, 5591, 5749, 5953, 6491, 6917, 7253, 8467, 9551, 9973, 10799, 11743, 12163, 12853, 15683, 16141, 19609, 31397, 34061, 35617, 35677

Offset

1,1

Comments

log(nextPrime(p))/log(p) is another measure of the (relative) gap between consecutive primes. See A144105 for the primes at the upper end of the gaps.

The statement log(prime(k+1))/log(prime(k)) < 1 + 1/k, for k >= 1, is a rewrite of the Firoozbakht conjecture. - John W. Nicholson, Dec 06 2013

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

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

Mathematica

nn=10^5; ps=N[Log[Prime[Range[nn]]]]; ps=Rest[ps]/Most[ps]; k=1; t={}; While[k<nn/2, mx=Max[Take[ps, {k, Length[ps]}]]; pos=Position[Take[ps, {k, Length[ps]}], mx][[ -1, 1]]; AppendTo[t, Prime[pos+k-1]]; k=k+pos]; t

Crossrefs

Sequence in context: A182047 A291544 A208149 * A088175 A271827 A298339

Adjacent sequences: A144101 A144102 A144103 * A144105 A144106 A144107

Keyword

nonn

Author

T. D. Noe, Sep 11 2008

Status

approved