A006880 -id:A006880 - cp's OEIS frontend

A006879 Number of primes with n digits.

0, 4, 21, 143, 1061, 8363, 68906, 586081, 5096876, 45086079, 404204977, 3663002302, 33489857205, 308457624821, 2858876213963, 26639628671867, 249393770611256, 2344318816620308, 22116397130086627, 209317712988603747, 1986761935284574233, 18906449883457813088, 180340017203297174362

Offset: 0

N. J. A. Sloane, Simon Plouffe

The number of primes between 10^(n-1) and 10^n. - Cino Hilliard, May 31 2008 [Corrected by Jon E. Schoenfield, Nov 29 2008]

			As 2, 3, 5, and 7 are the only primes less than 10, a(1) = 4.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 21, pp 8, Ellipses, Paris 2008.
C. T. Long, Elementary Introduction to Number Theory. Prentice-Hall, Englewood Cliffs, NJ, 1987, p. 77.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 179.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

Jianing Song, Table of n, a(n) for n = 0..29 (terms 0..24 by Charles R Greathouse IV, a(25) by Vladimir Pletser, a(26)-a(28) from David Baugh, a(29) based on A006880)
C. K. Caldwell, How Many Primes Are There?
Vladimir Pletser, Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems, Preprints.org, 2024. See p. 20.
Index entries for sequences related to numbers of primes in various ranges.

First differences of A006880.

Cf. A309329.

Differences[PrimePi[10^Range[-1, 25]]] (* Paolo Xausa, Apr 16 2024 *)

a(n)=primepi(10^n)-primepi(10^(n-1)) \\ Charles R Greathouse IV, May 03 2012

a(n) = pi(10^n)-pi(10^(n-1)) where pi(10^(-1)) := 0 (cf. A000720 and A006880).

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

a(11) and a(12) corrected by Jud McCranie and Enoch Haga
a(19) corrected and a(20) added by Paul Zimmermann
a(21)-a(22) from Vladeta Jovovic, Nov 07 2001

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

Original entry on oeis.org

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

A091634 Number of primes less than 10^n which do not contain the digit 0.

Original entry on oeis.org

4, 25, 153, 1010, 7122, 52313, 397866, 3103348, 24649318, 198536215, 1616808581, 13287264748, 110033428309, 917072930187

Offset: 1

Enoch Haga, Jan 30 2004

			a(3) = 153 because there are 168 primes less than 10^3, 15 primes have at least one zero; 168 - 15 = 153.

a(n) + A091644(n) = A006880(n).

Cf. A091635, A091636, A091637, A091638, A091639, A091640, A091641, A091642, A091643.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; c = 0; p = 1; Do[ While[ p = NextPrim[p]; p < 10^n, If[ Position[ IntegerDigits[p], 0] == {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)
Table[PrimePi[10^n]-Total[Boole[DigitCount[#,10,0]>0]&/@ Prime[ Range[ PrimePi[ 10^n]]]],{n,8}] (* The program generates the first 8 terms of the sequence. To generate more, increase the digit 8 but the program may take a long time to run. *) (* Harvey P. Dale, Aug 26 2021 *)

from sympy import sieve # use primerange for larger terms
def nodigs0(n): return '0' not in str(n)
def aupton(terms):
  ps, alst = 0, []
  for n in range(1, terms+1):
    ps += sum(nodigs0(p) for p in sieve.primerange(10**(n-1), 10**n))
    alst.append(ps)
  return alst
print(aupton(7)) # Michael S. Branicky, Apr 25 2021

Number of primes less than 10^n after removing any primes with at least one digit 0.
a(n) <= A052386(n) = 9*(9^n-1)/8. - Charles R Greathouse IV, Sep 13 2016
a(n) <= (9^n-1)/2 = A052386(n)*4/9 since the last digit of a prime of n digits can only be one of 4 numbers, (2,3,5,7) when n = 1 and (1,3,7,9) when n > 1. - Chai Wah Wu, Mar 18 2018

Edited and extended by Robert G. Wilson v, Feb 02 2004
a(9)-a(12) from Donovan Johnson, Feb 14 2008
a(13) from Robert Price, Nov 08 2013
a(14) from Giovanni Resta, Mar 20 2017

A091643 Number of primes less than 10^n which do not contain the digit 9.

Original entry on oeis.org

4, 19, 108, 687, 4766, 35139, 267486, 2083814, 16531372, 133059504, 1082995490, 8896945667, 73651718719, 613664827254

Offset: 1

Enoch Haga, Jan 30 2004

			a(2) = 19 because of the 25 primes less than 10^2, 6 have at least one digit 9; 25-6 = 19.

Cf. A091634, A091635, A091636, A091637, A091638, A091639, A091640, A091641, A091642.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; c = 0; p = 1; Do[ While[ p = NextPrim[p]; p < 10^n, If[ Position[ IntegerDigits[p], 9] == {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)

Number of primes less than 10^n after removing any primes with at least one digit 9.
a(n) = A006880(n) - A091710(n).
a(n) <= max(4,24*(9^(n-2))) <= 1 + (9^n)/3 (see formula in A091634). - Chai Wah Wu, Sep 17 2018

Edited and extended by Robert G. Wilson v, Feb 02 2004
a(9)-a(12) from Donovan Johnson, Feb 14 2008
a(13) from Robert Price, Nov 08 2013
a(14) from Giovanni Resta, Mar 20 2017

A073517 Number of primes less than 10^n with initial digit 1.

Original entry on oeis.org

0, 4, 25, 160, 1193, 9585, 80020, 686048, 6003530, 53378283, 480532488, 4369582734, 40063566855, 369893939287, 3435376839800, 32069022099022, 300694113015105, 2830466318006780, 26735673312004455, 253315661161665338, 2406763761677705769, 22923886160712831134, 218839439542390117580

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=4 because there are 4 primes up to 10^2 whose initial digit is 1 (11, 13, 17 and 19).

Chris K. Caldwell, How Many Primes Are There?
Xavier Gourdon & Pascal Sebah, Counting the number of primes
Henri Lifchitz, Parity of Pi(n)
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

Cf. A000720 (pi), A073509 to A073517, their sum is A006880.

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.

f[n_] := f[n] = PrimePi[2*10^n] - PrimePi[10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 13}]

a(n,d=1)=sum(k=0, n-1, primepi((d+1)*10^k-1) - primepi(d*10^k-1)) \\ Andrew Howroyd, Dec 15 2024

a(n) = Sum_{k=0..n-1} pi(2*10^k-1) - pi(10^k-1). - Andrew Howroyd, Dec 15 2024

Edited and extended by Robert G. Wilson v, Aug 29 2002
a(21)-a(22) added by David Baugh, Mar 21 2015
a(23) from Chai Wah Wu, Sep 18 2018
Offset corrected by Andrew Howroyd, Dec 15 2024

A073509 Number of primes less than 10^n with initial digit 9.

Original entry on oeis.org

0, 1, 15, 127, 1006, 8230, 70320, 614821, 5453140, 48982456, 444608278, 4070532710, 37535715441, 348245215460, 3247889171908, 30429496751905, 286235215995588, 2702000272361599, 25586688305447928, 242978340446949438, 2313264023790027111, 22074118786158858975

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2) = 1 because there is 1 prime less than 100 whose initial digit is 9, i.e., 97.

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

A006880(n) = A073509(n)+ ... + A073516(n)+A073517(n-1).

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509

f[n_] := f[n] = PrimePi[10^(n + 1)] - PrimePi[9*10^n] + f[n - 1]; f[0] = 0; Table[f[n], {n, 0, 12}]

Edited and extended by Robert G. Wilson v, Aug 29 2002

a(20)-a(22) added by David Baugh, Mar 22 2015

A073510 Number of primes less than 10^n with initial digit 8.

Original entry on oeis.org

0, 2, 17, 127, 1003, 8326, 71038, 618610, 5481646, 49221187, 446590932, 4087194991, 37677478288, 349465615584, 3258501713644, 30522628848972, 287059041039078, 2709339704446862, 25652489700275636, 243571629996128384, 2318640708958531064, 22123070798400775157

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=2 because there are 2 primes up to 10^2 whose initial digit is 8 (namely 83 and 89).

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

A006880(n) = A073509(n)+ ... + A073516(n)+A073517(n-1).

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509

f[n_] := f[n] = PrimePi[9*10^n] - PrimePi[8*10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 12}]

Edited and extended by Robert G. Wilson v, Aug 29 2002

a(20)-a(22) added by David Baugh, Mar 22 2015

A073511 Number of primes less than 10^n with initial digit 7.

Original entry on oeis.org

1, 4, 18, 125, 1027, 8435, 71564, 622882, 5516130, 49495432, 448855139, 4106164356, 37838546363, 350849788546, 3270531245684, 30628143485953, 287992070079777, 2717649138419586, 25726964404879666, 244242934202964444, 2324722877951987037, 22178433287546997612

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=4 because there are 4 primes up to 10^2 whose initial digit is 7 (namely 7, 71, 73 and 79).

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

Cf. A073509 to A073517, their sum is A006880.

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509

f[n_] := f[n] = PrimePi[8*10^n] - PrimePi[7*10^n] + f[n - 1]; f[0] = 1; Table[ f[n], {n, 0, 12}]

Edited and extended by Robert G. Wilson v, Aug 29 2002

a(20)-a(22) added by David Baugh, Mar 22 2015

A073512 Number of primes less than 10^n with initial digit 6.

Original entry on oeis.org

0, 2, 18, 135, 1013, 8458, 72257, 628206, 5556434, 49815418, 451476802, 4128049326, 38024311091, 352446754137, 3284400373590, 30749731897370, 289066731934716, 2727216210298152, 25812680778645432, 245015325044029789, 2331718909954888809, 22242097596092999144

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=2 because there are 2 primes up to 10^2 whose initial digit is 2 (namely 61 and 67).

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

Cf. A073509 to A073517, their sum is A006880.

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509

f[n_] := f[n] = PrimePi[7*10^n] - PrimePi[6*10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 12}]

Edited and extended by Robert G. Wilson v, Aug 29 2002

a(20)-a(22) added by David Baugh, Mar 22 2015

A073513 Number of primes less than 10^n with initial digit 5.

Original entry on oeis.org

1, 3, 17, 131, 1055, 8615, 72951, 633932, 5602768, 50193913, 454577490, 4153943134, 38243708524, 354330372215, 3300752009165, 30892997367352, 290332329192655, 2738477783884855, 25913537508233527, 245923809778144431, 2339944887042508496, 22316931815316988517

Offset: 1

Shyam Sunder Gupta, Aug 14 2002

			a(2)=3 because there are 3 primes up to 10^2 whose initial digit is 5 (namely 5, 53 and 59).

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

Cf. A073509 to A073517, their sum is A006880.

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509

f[n_] := f[n] = PrimePi[6*10^n] - PrimePi[5*10^n] + f[n - 1]; f[0] = 1; Table[ f[n], {n, 0, 13}]

Edited and extended by Robert G. Wilson v, Aug 29 2002

a(20)-a(22) added by David Baugh, Mar 22 2015

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A006879 Number of primes with n digits.

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

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A091634 Number of primes less than 10^n which do not contain the digit 0.

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A091643 Number of primes less than 10^n which do not contain the digit 9.

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A073517 Number of primes less than 10^n with initial digit 1.

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A073509 Number of primes less than 10^n with initial digit 9.

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A073510 Number of primes less than 10^n with initial digit 8.

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