A274767 -id:A274767 - cp's OEIS frontend

A006988 a(n) = (10^n)-th prime.

2, 29, 541, 7919, 104729, 1299709, 15485863, 179424673, 2038074743, 22801763489, 252097800623, 2760727302517, 29996224275833, 323780508946331, 3475385758524527, 37124508045065437, 394906913903735329, 4185296581467695669, 44211790234832169331

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Check the b-file for terms beyond those listed above.

			a(0) = 10^0-th prime = first prime = 2.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 111.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Marc Deleglise et al., Table of n, a(n) for n = 0..24 (a(23) corrected and a(24) added using Kim Walisch's primecount program, by David Baugh, Nov 11 2015)
C. K. Caldwell, Marc Deleglise's work on new values of pi(x)
UTM, The Nth Prime Page.
Eric Weisstein's World of Mathematics, Prime Number
R. G. Wilson, V, Letter to N. J. A. Sloane, Jan. 1989

Cf. A006880, A033844, A119780.

Cf. A099260, A274767 ((leading) digits of 103-digit a(100)).

A006988:=n->ithprime(10^n): seq(A006988(n), n=0..7); # Wesley Ivan Hurt, Nov 21 2014

Mathematica
```
Table[Prime[10^n], {n, 0, 12}]
```

a(n)=prime(10^n) \\ Charles R Greathouse IV, Jul 21 2011

More terms from Paul Zimmermann
a(19) from Marc Deleglise, Jun 29 2008
a(20) found by Andrey V. Kulsha using a program by Xavier Gourdon, Oct 05 2011
a(21) from Henri Lifchitz, Sep 09 2014
a(22) from Henri Lifchitz, Nov 21 2014

A121046 Approximation to the (10^n)-th prime by applying a bisection to Gram's formula for Riemann's approximation of the prime counting function.

Original entry on oeis.org

29, 536, 7923, 104768, 1299733, 15484040, 179431239, 2038076587, 22801797576, 252097715777, 2760727752353, 29996225393465, 323780512411510, 3475385760290723, 37124508056355511, 394906913798224975, 4185296581676470068, 44211790234127235470

Offset: 1

Cino Hilliard, Aug 08 2006, Aug 17 2006

nonn

The algorithm primex(x) uses an exponent bisection routine and Gram's Riemann approximation, Rg(x) for the prime counting function pi(x). We know that Rg(x) is relatively close to pi(x) as x gets large. We take advantage of this relatively small error noting that pi(prime(x)) = x ~ Rg(prime(x)). A reasonable approximation of prime(x) is x*log(x) while for x = 10^n, often, 10^n*log(10^(n+1)) is a much better approximation. The PARI program shows the flow of this algorithm.

			pi(10^18) = A006988(18) = 44211790234832169331 and a(18) = 44211790234127235470. So the approximation of pi(10^18) by primex(10^18) is accurate to 11 places.
Agrees for 52 digits with the solution to Li(x)=10^100 given in Mathematics Stack Exchange link. - _Hugo Pfoertner_, Nov 17 2019

Hugo Pfoertner, Table of n, a(n) for n = 1..100
Chris Caldwell, The Prime Page.
Cino Hilliard, David Broadhurst, Andrey Kulsha, Number of prime-index-primes < n, digest of 15 messages in primeform Yahoo group, Apr 2 - Apr 5, 2006. [Cached copy]
Mathematics Stack Exchange, How many digits of the googol-th prime can we calculate (or were calculated)?

Cf. A006988, A052434, A274767.

\\ List the approximations to the (10^n)-th prime by Cino Hilliard
\\ Gram's Riemann's Approx of Pi(x)
Rg(x) = { local(n=1, L, s=1, r); L=r=log(x); while(s<10^120*r, s=s+r/zeta(n+1)/n; n=n+1; r=r*L/n); (s) }
primex(n) = { local(x, px, r1, r2, r, p10, b, e); b=10; p10=log(n)/log(10); if(Rg(b^p10*log(b^(p10+1)))< b^p10, m=p10+1, m=p10); r1 = 0; r2 = 1; for(x=1, 400, r=(r1+r2)/2; px = Rg(b^p10*log(b^(m+r))); if(px <= b^p10, r1=r, r2=r); r=(r1+r2)/2; ); floor(b^p10*log(b^(m+r))+.5); }
for (k=1,20,print1(primex(10^k),", "))

More terms from Hugo Pfoertner, Nov 17 2019

More precise name by Hugo Pfoertner, Apr 29 2021

cp's OEIS Frontend

A006988 a(n) = (10^n)-th prime.

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