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Length in bits of (10^n)-th prime number.
1

%I #4 Mar 30 2012 17:36:43

%S 2,5,10,13,17,21,24,28,31,35,38,42,45,49,52,56,59,62,66,69,73,76,79,

%T 83,86,89,93,96,100,103,106,110,113,116,120,123,127,130,133,137,140,

%U 143,147,150,153,157,160,163,167,170,173,177,180,184,187,190,194,197,200,204

%N Length in bits of (10^n)-th prime number.

%H Pierre Dusart, <a href="http://arxiv.org/abs/1002.0442">Estimates of Some Functions Over Primes Without R.H.</a>

%e a(1) = 5 because A006988(1) = prime(10^1) = 29 = 11101 (base 2) has five bits.

%o (PARI) a(n)=if(n<3,return([2,5,10][n+1]));my(l=n*log(10),ll=log(l),x=n*log(10)/log(2),lb=ceil(x+log(l+ll-1+(ll-2.2)/l)/log(2)),ub=ceil(x+log(l+ll-1+(ll-2)/l)/log(2)));if(lb==ub,lb,error("Cannot determine a("n")"))

%Y Cf. A006988 ((10^n)-th prime), A006880 (pi(10^n)), A007053 (pi(2^n)), A099260 (decimal digit lengths).

%K nonn,base

%O 0,1

%A _Rick L. Shepherd_, Oct 11 2004

%E Extension, program, and reference from _Charles R Greathouse IV_, Aug 03 2010