A082886 - OEIS

%I #6 Sep 04 2015 10:44:23

%S 1,1,1,2,0,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0,0,2,0,1,0,2,

%T 0,1,1,0,1,1,0,1,0,0,0,2,2,0,0,0,1,0,1,1,1,1,0,1,0,0,1,2,0,0,0,2,1,1,

%U 0,0,1,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,0,1,1,0,1,0,0,1,0,2,0,1,0,0,0,0

%N floor((prime(n+1)-prime(n))/log(prime(n))).

%C a(n) is unbounded by a theorem of Westzynthius. - _Charles R Greathouse IV_, Sep 04 2015

%H Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao, <a href="http://arxiv.org/abs/1412.5029">Long gaps between primes</a> (2014).

%F a(n)=floor((prime(n+1)-prime(n))/log(prime(n))).

%F a(n)=Floor(A001223(n)/log(A000040(n))).

%F Infinitely often a(n) >> log log n log log log log n/log log log n, see Ford-Green-Konyagin-Maynard-Tao. - _Charles R Greathouse IV_, Sep 04 2015

%e a(217) = floor((1361-1327)/log(1327)) = floor(4.72834...) = 4.

%t Table[Floor[(Prime[n+1]-Prime[n])/Log[Prime[n]]//N], {n, 1, 220}]

%o (PARI) a(n,p=prime(n))=(nextprime(p+1)-p)\log(p) \\ _Charles R Greathouse IV_, Sep 04 2015

%Y Cf. A082862, A082884, A082885, A082888-A082891.

%K nonn

%O 1,4

%A _Labos Elemer_, Apr 17 2003