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%I #8 Dec 12 2020 17:09:55
%S 1,2,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,45,46,47,48,49,50,
%U 51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75
%N Number of decimal digits in (10^n)-th prime number.
%C As lim {n->oo} p_n/(n log n) = 1 is equivalent to the prime number theorem, a good first approximation (without having done any detailed analysis) should be a(n)=floor(log_10((10^n)*log(10^n))), which correctly generates all the first 22 terms and predicts that the sequence will continue 24,25,...,43,44,46,47,...,435,436,438,439,...,4344,4345,4347,4348,...,4503,4504 through the first 4500 terms (with only 5,45,437,4346 not appearing - compare with the digits of log_10(e) in A002285).
%C Many terms of this sequence can be determined exactly using Dusart's bounds. The first missing terms are 5, 44, 435, 4344, 43430, 434295, 4342946, 43429449, 434294483, 4342944820, ....
%H Pierre Dusart, <a href="http://arxiv.org/abs/1002.0442">Estimates of Some Functions Over Primes without R.H.</a>
%e a(4) = 6 because A006988(4) = prime(10^4) = 104729 has six decimal digits.
%t Table[IntegerLength[Prime[10^n]],{n,0,75}] (* _Harvey P. Dale_, Dec 11 2020 *)
%o (PARI) a(n)=if(n<3,return(n+1));my(l=n*log(10),ll=log(l),lb=ceil(log(l+ll-1+(ll-2.2)/l)/log(10)),ub=ceil(log(l+ll-1+(ll-2)/l)/log(10)));if(lb==ub,n+lb,error("Cannot determine a("n")"))
%Y Cf. A006988 ((10^n)-th prime), A006880 (pi(10^n)), A099261 (bit lengths).
%K nonn,base,nice
%O 0,2
%A _Rick L. Shepherd_, Oct 10 2004
%E Extension, comment, link, and Pari program from _Charles R Greathouse IV_, Aug 03 2010