A082884 a(n) = smallest prime p for which (r-p)/log(p) < 1/n, where r is the next prime after p.

11, 59, 419, 2999, 22037, 162821, 1202627, 8886329, 65660051, 485165279, 3584913989, 26489122349, 195729610331, 1446257064389, 10686474581831, 78962960185097, 583461742527491, 4311231547116551, 31855931757115889

Offset: 1

Author

Labos Elemer, Apr 17 2003

Keywords

• nonn

Comments

Remark: the quotient can be larger than 1/n at much larger primes, many times. So p does not set record in "standard" sense, only sinks first below 1/n, i.e. smaller than any previous values, but not necessarily smaller than the following ones. See also illustration.

Examples

			a(1)=p(5)=11 and (13-11)/log(11) = 0.8340... < 1/1.
		

Crossrefs

Cf. A082862, A082885, A082886, A082888-A082891.

Programs

Mathematica

{eq=1, k=1}; Do[s=(Prime[n+1]-Prime[n])/Log[Prime[n]]//N; If[s<1/k, k=k+1; Print[{k, n, Prime[n], s}]; eq=s], {n, 1, 100000000}]

Formula

a(n)=min{prime(j): A001223(j)/log(prime(j)) < 1/n}, where prime(j)=A000040(j) is the j-th prime.

Extensions

a(11)-a(19) from Donovan Johnson, Sep 09 2008