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Primes p such that log(nextPrime(p))/log(p) is smaller for larger primes.
4

%I #34 Mar 17 2018 23:04:26

%S 2,3,7,13,23,31,47,53,113,139,199,211,293,317,523,1327,1669,1951,2179,

%T 2477,2971,3271,4297,4831,5591,5749,5953,6491,6917,7253,8467,9551,

%U 9973,10799,11743,12163,12853,15683,16141,19609,31397,34061,35617,35677

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

%C 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.

%C 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

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

%C 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

%H T. D. Noe, <a href="/A144104/b144104.txt">Table of n, a(n) for n = 1..176</a>

%H A. Kourbatov, <a href="http://arxiv.org/abs/1503.01744">Verification of the Firoozbakht conjecture for primes up to four quintillion</a>, arXiv:1503.01744 [math.NT], 2015

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Firoozbakht%E2%80%99s_conjecture">Firoozbakht's conjecture</a>

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

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

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

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

%e (log(35729)/log(35677))^3795 = 1.69... < (1+1/3795)^3795 = 2.717... < e. - _Daniel Forgues_, Apr 28 2014

%t 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

%K nonn

%O 1,1

%A _T. D. Noe_, Sep 11 2008