A006988 -id:A006988 - cp's OEIS frontend

A006880 Number of primes < 10^n.

0, 4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 455052511, 4118054813, 37607912018, 346065536839, 3204941750802, 29844570422669, 279238341033925, 2623557157654233, 24739954287740860, 234057667276344607, 2220819602560918840, 21127269486018731928, 201467286689315906290

Offset: 0

N. J. A. Sloane and Simon Plouffe

Number of primes with at most n digits; or pi(10^n).
Partial sums of A006879. - Lekraj Beedassy, Jun 25 2004
Also omega( (10^n)! ), where omega(x): number of distinct prime divisors of x. - Cino Hilliard, Jul 04 2007
This sequence also gives a good approximation for the sum of primes less than 10^(n/2). This is evident from the fact that the number of primes less than 10^2n closely approximates the sum of primes less than 10^n. See link on Sum of Primes for the derivation. - Cino Hilliard, Jun 08 2008
It appears that (10^n)/log((n+3)!) is a lower bound close to a(n), see A025201. - Eric Desbiaux, Jul 20 2010, edited by M. F. Hasler, Dec 03 2018

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 143, 146.
Richard Crandall and Carl B. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; p. 11.
Keith Devlin, Mathematics: The New Golden Age, new and revised edition. New York: Columbia University Press (1993): p. 6, Table 1.
Marcus du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; p. 48.
Calvin T. Long, Elementary Introduction to Number Theory. Prentice-Hall, Englewood Cliffs, NJ, 1987, p. 77.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 179.
H. Riesel, "Prime numbers and computer methods for factorization," Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, page 38.
D. Shanks, Solved and Unsolved Problems in Number Theory. Chelsea, NY, 2nd edition, 1978, p. 15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 455052511 at p. 190.

David Baugh, Table of n, a(n) for n = 0..29 (terms n = 1..27 from Charles R Greathouse IV).
Chris K. Caldwell, How Many Primes Are There?
Chris K. Caldwell, Mark Deleglise's work
Muhammed Hüsrev Cilasun, An Analytical Approach to Exponent-Restricted Multiple Counting Sequences, arXiv preprint arXiv:1412.3265 [math.NT], 2014.
Muhammed Hüsrev Cilasun, Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3.
Jens Franke, Thorsten Kleinjung, Jan Büthe, and Alexander Jost, A practical analytic method for calculating pi(x), Math. Comp. 86 (2017), 2889-2909.
Xavier Gourdon, a(22) found by pi(x) project
Xavier Gourdon and Pascal Sebah, The pi(x) project : results and current computations
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Cino Hilliard, Sum of primes [unusable link]
Ronald K. Hoeflin, Titan Test
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979, [see p. 6 of the pdf].
Rishi Kumar, Kepler Sets of Second-Order Linear Recurrence Sequences Over Q_p, arXiv:2406.05890 [math.NT], 2024. See p. 7.
J. C. Lagarias, V. S. Miller, and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., 44 (1985), pp. 537-560.
J. C. Lagarias and Andrew M. Odlyzko, Computing pi(x): An analytic method, J. Algorithms, 8 (1987), pp. 173-191.
Pieter Moree, Izabela Petrykiewicz, and Alisa Sedunova, A computational history of prime numbers and Riemann zeros, arXiv:1810.05244 [math.NT], 2018. See Table 1 p. 6.
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
Tomás Oliveira e Silva, Computing pi(x): the combinatorial method, Revista do Detua, Vol. 4, N 6, March 2006.
David J. Platt, Computing pi(x) analytically, arXiv:1203.5712 [math.NT], 2012-2013.
Vladimir Pletser, Conjecture on the value of Pi(10^26), the number of primes less than 10^26, arXiv:1307.4444 [math.NT], 2013.
Vladimir Pletser, Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems, Preprints.org, 2024. See p. 20.
Douglas B. Staple, The combinatorial algorithm for computing pi(x), arXiv:1503.01839 [math.NT], 2015.
M. R. Watkins, The distribution of prime numbers
Eric Weisstein's World of Mathematics, Prime Counting Function
Wikipedia, Prime number theorem
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1989
Index entries for sequences related to numbers of primes in various ranges

Cf. A000720, A006879, A006988, A007053, A011557, A025201, A040014.

a006880 = a000720 . (10 ^)  -- Reinhard Zumkeller, Mar 17 2015

Table[PrimePi[10^n], {n, 0, 14}] (* Jean-François Alcover, Nov 08 2016, corrected Sep 29 2020, a(14) being the maximum computable with certain implementations *)

a(n)=primepi(10^n) \\ Charles R Greathouse IV, Nov 08 2011

a(n) = A000720(10^n). - M. F. Hasler, Dec 03 2018

Limit_{n->oo} a(n)/a(n-1) = 10. - Stefano Spezia, Aug 31 2025

Lehmer gave the incorrect value 455052512 for the 10th term. More terms May 1996. Jud McCranie points out that the 11th term is not 4188054813 but rather 4118054813.
a(22) from Robert G. Wilson v, Sep 04 2001
a(23) (see Gourdon and Sebah) has yet to be verified and the assumed error is +-1. - Robert G. Wilson v, Jul 10 2002 [The actual error was 14037804. - N. J. A. Sloane, Nov 28 2007]
a(23) corrected by N. J. A. Sloane from the web page of Tomás Oliveira e Silva, Nov 28 2007
a(25) from J. Buethe, J. Franke, A. Jost, T. Kleinjung, Jun 01 2013, who said: "We have calculated pi(10^25) = 176846309399143769411680 unconditionally, using an analytic method based on Weil's explicit formula".
a(26) from Douglas B. Staple, Dec 02 2014
a(27) in the b-file from David Baugh and Kim Walisch via Charles R Greathouse IV, Jun 01 2016
a(28) in the b-file from David Baugh and Kim Walisch, Oct 26 2020
a(29) in the b-file from David Baugh and Kim Walisch, Feb 28 2022

A033844 a(n) = prime(2^n).

Original entry on oeis.org

2, 3, 7, 19, 53, 131, 311, 719, 1619, 3671, 8161, 17863, 38873, 84017, 180503, 386093, 821641, 1742537, 3681131, 7754077, 16290047, 34136029, 71378569, 148948139, 310248241, 645155197, 1339484197, 2777105129, 5750079047, 11891268401, 24563311309, 50685770167, 104484802057, 215187847711

Offset: 0

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

a(n) is the smallest number m such that pi(m)=d(m)^n, where d(m) is number of positive divisors of m (the proof is easy). - Farideh Firoozbakht, Jun 06 2005

David Baugh, Table of n, a(n) for n = 0..78 (terms n = 0..57 from Charles R Greathouse IV, terms n = 58..78 found using Kim Walisch's primecount program).

Cf. A000040, A018249, A051438, A051440, A051439, A006988, A107655.

Mathematica
```
Table[Prime[2^n], {n, 0, 32}]
```

a(n)=prime(2^n) \\ Charles R Greathouse IV, Nov 02 2014

More terms from Robert G. Wilson v, Jun 09 2000

A119290 a(n) is the total number of digits in the first 10^n primes.

Original entry on oeis.org

1, 16, 271, 3803, 48982, 610484, 7245905, 83484450, 942636916, 10487584405, 115369529592, 1257761617574, 13611696080735, 146406754329933, 1566562183907264, 16687323842873339, 177063766685219106, 1872323812397478246, 19738266145121133639, 207517446542560214799, 2176390177056541482871, 22774922890367225576581

Offset: 0

Enoch Haga, May 13 2006

			At a(1) there are 10^1 primes, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and the total number of digits is 16.

Amiram Eldar, Table of n, a(n) for n = 0..24

Cf. A006880, A006988, A097952.

Accumulate@Table[c = 0; i0 = If[n == 0, 1, 10^(n - 1) + 1]; For[i = i0, i <= 10^n, i++, c += IntegerLength[Prime[i]]]; c, {n, 0, 6}] (* Robert Price, Jun 09 2019 *)

Count the digits in the first 10^n primes.

a(n) = sum while positive from k=0 to (10^n - A006880(k)). - Charles R Greathouse IV, Jul 09 2007

Corrected and extended by Charles R Greathouse IV, Jul 09 2007

A038833 3^n-th prime.

Original entry on oeis.org

2, 5, 23, 103, 419, 1543, 5519, 19289, 65687, 220861, 733561, 2412779, 7867547, 25485211, 82064027, 262960091, 839001721, 2666790167, 8448283757, 26684834641, 84064182061, 264194301347, 828513359549, 2593131866483, 8101615860589, 25270000074757, 78701296245541, 244766272211729, 760254097357847, 2358539553316997

Offset: 0

Jeff Burch

Charles R Greathouse IV, Table of n, a(n) for n = 0..36

Cf. A033844, A006988.

Mathematica
```
Table[Prime[3^n], {n, 0, 23}]
```

a(n)=prime(3^n) \\ Charles R Greathouse IV, Nov 02 2014

a(24)-a(29) from Charles R Greathouse IV, Nov 02 2014

A099825 Sum of the first 2^n primes.

Original entry on oeis.org

2, 5, 17, 77, 381, 1851, 8893, 41741, 191755, 868151, 3875933, 17120309, 74950547, 325590115, 1405167561, 6029676711, 25750781177, 109495928099, 463852117169, 1958476902435, 8244703036797, 34615624751259, 144991244981985, 605994279458465, 2527803622205465

Offset: 0

Robert G. Wilson v, Oct 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..65 (calculated using Kim Walisch's primesum program; terms 0..30 from Robert G. Wilson v)
Cino Hilliard, Sumprimes. [broken link]
Kim Walisch, Sum of the primes below x (primesum).

Cf. A000040, A000079, A006988, A007504, A033844, A099824, A099826, A121248.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[0] = 2; f[n_] := f[n] = Block[{k = 0, mx = 2^n/2, np = Prime[2^n/2], s = f[n - 1]}, While[k < mx, k++; np = NextPrim@np; s = s + np]; s]; Table[ f@n, {n, 0, 23}] (* Robert G. Wilson v, Aug 24 2006 *)
Module[{nn=22,ap},ap=Accumulate[Prime[Range[2^nn]]];Table[ap[[2^n]],{n,0,nn}]] (* Harvey P. Dale, Apr 12 2017 *)

a(n)=my(s); n=2^n; forprime(p=2,, s+=p; if(n--==0, return(s))) \\ Charles R Greathouse IV, Feb 16 2017 \\ corrected by David A. Corneth, Aug 05 2025

a(n) = A007504(A000079(n)). - Amiram Eldar, Jul 01 2024

A099824 a(n) = Sum of the first 10^n primes.

Original entry on oeis.org

2, 129, 24133, 3682913, 496165411, 62260698721, 7472966967499, 870530414842019, 99262851056183695, 11138479445180240497, 1234379338586942892505, 135436174616790289414111, 14738971133550183905879827, 1593061976858155930556059673, 171191473337951767580578821529

Offset: 0

Robert G. Wilson v, Oct 25 2004

nonn

			For n=1, the sum of the first 10^1 = 10 primes is 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129, so a(1) = 129. - _Michael B. Porter_, Aug 08 2016

David Baugh, Table of n, a(n) for n = 0..23 [terms a(0)-a(17) from Marc Deleglise; terms a(18)-a(21) from Kim Walisch]
Cino Hilliard, SumPrimes, June 2008. [broken link]

Cf. A000040, A006988, A007504, A099825, A099826.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; k = p = 1; s = 0; Do[ While[p = NextPrim[p]; s = s + p; k < 10^n, k++ ]; k++; Print[s], {n, 0, 8}]
Table[Sum[Prime[i], {i, 10^n}], {n, 0, 5}] (* José de Jesús Camacho Medina, Dec 27 2014 *)

vecA099824(n)={ my(s,c,k=1,L:list); L=List();
forprime(m=2,prime(10^n),s+=m;c++; if(c==k,listput(L,s);k*=10));
return(vector(#L,i,L[i]))} \\ R. J. Cano, Aug 12 2016

a(n) = Sum_{i=1..10^n} A000040(i). - José de Jesús Camacho Medina, Dec 27 2014 (corrected by Joerg Arndt, Jan 05 2015)

a(9) from Hans Havermann, May 06 2005
a(10) from Cino Hilliard, Apr 28 2006
a(11) from Cino Hilliard, Oct 03 2006
a(12)-a(13) from Hiroaki Yamanouchi, Jul 06 2014
a(11) corrected by Marc Deleglise, Apr 03 2016
a(14)-a(17) from Marc Deleglise, Apr 03 2016
a(18)-a(20) from Kim Walisch, Jun 05 2016
a(21) from Kim Walisch, Jun 11 2016
a(22) from David Baugh using Kim Walisch's primesum program, Jun 21 2016
a(23) from David Baugh using Kim Walisch's primesum program, Sep 26 2016

A099826 Sum of the first 3^n primes.

Original entry on oeis.org

2, 10, 100, 1264, 15116, 171148, 1864190, 19697700, 203534530, 2067129306, 20706364528, 205144046742, 2014349179358, 19632546354498, 190150622868298, 1831906588192414, 17567504017456404, 167794196312059488, 1597037992049539274

Offset: 0

Robert G. Wilson v, Oct 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..36 (calculated using Kim Walisch's primesum program)
Cino Hilliard, Sum first 2^n Primes. [Link corrected by Alexey Beshenov, Jan 25 2009] [broken link]
Kim Walisch, Sum of the primes below x (primesum).

Cf. A000040, A000244, A006988, A007504, A038833, A099824, A099825.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; k = p = 1; s = 0; Do[ While[p = NextPrim[p]; s = s + p; k < 10^n, k++ ]; k++; Print[s], {n, 0, 16}]

a(n) = A007504(A000244(n)). - Amiram Eldar, Jul 01 2024

More terms from Cino Hilliard, Jan 14 2006

A114125 a(n) is the 10^n-th semiprime.

Original entry on oeis.org

4, 26, 314, 3595, 40882, 459577, 5109839, 56168169, 611720495, 6609454805, 70937808071, 757060825018, 8040423200947, 85037651263063, 896113850117314, 9413000361625346, 98597629032410971, 1030179406403917981, 10739422018595513973, 111729397883168684917, 1160260967837159869621

Offset: 0

Robert G. Wilson v, Feb 11 2006

nonn

Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A006880, A006988, A066265, A078972.

fQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; c = 0; k = 2; Do[While[c < 10^n, If[fQ@k, c++ ]; k++ ]; Print[k - 1], {n, 0, 8}]
(* checked by *) SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]

use ntheory ":all"; print "$ ",nth_semiprime(10**$),"\n" for 0..15; # Dana Jacobsen, Oct 08 2018

a(14) from Donovan Johnson, Sep 27 2010
Corrected a(14), added a(15)-a(18) from Dana Jacobsen, Oct 10 2018
a(19)-a(20) from Henri Lifchitz, Nov 08 2024

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

A006880 Number of primes < 10^n.

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A033844 a(n) = prime(2^n).

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A119290 a(n) is the total number of digits in the first 10^n primes.

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A038833 3^n-th prime.

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A099825 Sum of the first 2^n primes.

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A099824 a(n) = Sum of the first 10^n primes.

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A099826 Sum of the first 3^n primes.

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A114125 a(n) is the 10^n-th semiprime.