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A099260 Number of decimal digits in (10^n)-th prime number. 1
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, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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, ....

LINKS

Table of n, a(n) for n=0..72.

Pierre Dusart, Estimates of Some Functions Over Primes without R.H.

EXAMPLE

a(4) = 6 because A006988(4) = prime(10^4) = 104729 has six decimal digits.

PROG

(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")"))

CROSSREFS

Cf. A006988 ((10^n)-th prime), A006880 (pi(10^n)), A099261 (bit lengths).

Sequence in context: A071789 A131870 A004724 * A231237 A053241 A132329

Adjacent sequences:  A099257 A099258 A099259 * A099261 A099262 A099263

KEYWORD

nonn,base,nice

AUTHOR

Rick L. Shepherd, Oct 10 2004

EXTENSIONS

Extension, comment, link, and Pari program from Charles R Greathouse IV, Aug 03 2010

STATUS

approved

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Last modified December 22 05:04 EST 2014. Contains 252328 sequences.