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A082884 a(n) = smallest prime p for which (r-p)/log(p) < 1/n, where r is the next prime after p. 8


%S 11,59,419,2999,22037,162821,1202627,8886329,65660051,485165279,

%T 3584913989,26489122349,195729610331,1446257064389,10686474581831,

%U 78962960185097,583461742527491,4311231547116551,31855931757115889

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

%C 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.

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

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

%t {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}]

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

%K nonn

%O 1,1

%A _Labos Elemer_, Apr 17 2003

%E a(11)-a(19) from _Donovan Johnson_, Sep 09 2008

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Last modified July 5 23:36 EDT 2020. Contains 335475 sequences. (Running on oeis4.)