cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

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

A091645 Number of primes less than 10^n having at least one digit 1.

Original entry on oeis.org

0, 8, 67, 559, 4917, 44073, 402030, 3709989, 34534791, 323587790, 3046685950, 28798331502, 273078628285, 2596399340798
Offset: 1

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Examples

			a(2) = 8 because of the 25 primes less than 10^2, 8 have at least one 1 digit.

Crossrefs

a(n) + A091635(n) = A006880(n).
Cf. A091645, A091646, A091647, A091705, A091706, A091707, A091708, A091709, A091710.

Programs

  • Mathematica
    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], 1] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)
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], 2] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)

Extensions

Edited, corrected and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 20 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091644 Number of primes less than 10^n which have at least one digit 0.

Original entry on oeis.org

0, 0, 15, 219, 2470, 26185, 266713, 2658107, 26198216, 256516296, 2501246232, 24320647270, 236032108530, 2287868820615
Offset: 1

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Comments

3 additional terms, generated using a sieve program. - Ryan Propper, Aug 20 2005

Examples

			a(3) = 15 because of the 168 primes less than 10^3, 15 have at least one 0 digit.

Crossrefs

Cf. A091645, A091646, A091647, A091705, A091706, A091707, A091708, A091709, A091710.

Programs

  • Mathematica
    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 *)
  • Python
    from sympy import sieve # use primerange for larger terms
def digs0(n): return '0' in str(n)
def aupton(terms):
  ps, alst = 0, []
  for n in range(1, terms+1):
    ps += sum(digs0(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

Formula

a(n) = A006880(n) - A091634(n).

Extensions

Edited and extended by Robert G. Wilson v, Feb 02 2004
More terms from Ryan Propper, Aug 20 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091646 Number of primes less than 10^n having at least one digit 2.

Original entry on oeis.org

1, 3, 29, 352, 3357, 32393, 312424, 3014171, 29016211, 279171099, 2685272908, 25829666453, 248507003625, 2391688694305
Offset: 1

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Examples

			a(2) = 3 because of the 25 primes less than 10^2, 3 have at least one digit 2.

Crossrefs

a(n) + A091636(n) = A006880(n).
Cf. A091644, A091645, A091647, A091705, A091706, A091707, A091708, A091709, A091710.

Extensions

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 20 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091647 Number of primes less than 10^n having at least one digit 3.

Original entry on oeis.org

1, 9, 66, 561, 4877, 43685, 399564, 3694303, 34433999, 322852288, 3041362298, 28758431735, 272777483044, 2594081699837
Offset: 1

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Examples

			a(2) = 9 because of the 25 primes less than 10^2, 9 have at least one digit 3.

Crossrefs

a(n) + A091637(n) = A006880(n).
Cf. A091637, A091644, A091645, A091646, A091705, A091706, A091707, A091708, A091709, A091710.

Programs

  • Mathematica
    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], 3] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)

Extensions

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 20 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091707 Number of primes less than 10^n having at least one digit 6.

Original entry on oeis.org

0, 2, 32, 332, 3225, 31792, 309431, 2991216, 28863327, 278085918, 2677289604, 25769815303, 248050788931, 2388172543971
Offset: 1

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Examples

			a(1) = 0 because of the 4 primes less than 10^1, none have at least one digit 6.

Crossrefs

a(n) + A091640(n) = A006880(n).
Cf. A091644, A091645, A091646, A091647, A091705, A091706, A091708, A091709, A091710.

Programs

  • Mathematica
    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], 6] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)
  • Python
    from sympy import sieve # use slower primerange for larger terms
def a(n): return sum('6' in str(p) for p in sieve.primerange(2, 10**n))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Apr 23 2021

Extensions

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 21 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091708 Number of primes less than 10^n having at least one digit 7.

Original entry on oeis.org

1, 9, 68, 549, 4819, 43506, 397756, 3681943, 34344296, 322199867, 3036544958, 28722439343, 272501681533, 2591959274190
Offset: 1

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Examples

			a(1) = 1 because of the 4 primes less than 10^1, 1 has at least one digit 7.

Crossrefs

a(n) + A091641(n) = A006880(n).
Cf. A091644, A091645, A091646, A091647, A090705, A090706, A091707, A091709, A091710.

Programs

  • Mathematica
    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], 7] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)
Table[Count[Prime[Range[PrimePi[10^n]]],?(DigitCount[#,10,7]>0&)],{n,12}] (* _Harvey P. Dale, Mar 03 2013 *)

Extensions

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 21 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091709 Number of primes less than 10^n having at least one digit 8.

Original entry on oeis.org

0, 2, 27, 314, 3217, 31699, 308774, 2987107, 28824402, 277779084, 2674980022, 25752370493, 247919235555, 2387154761520
Offset: 1

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Examples

			a(2) = 2 because of the 25 primes less than 10^2, 2 have at least one digit 8.

Crossrefs

a(n) + A091642(n) = A006880(n).
Cf. A091644, A091645, A091646, A091647, A090705, A090706, A091707, A091708, A091710.

Programs

  • Mathematica
    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], 8] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)
  • Python
    from sympy import sieve # use slower primerange for larger terms
def a(n): return sum('8' in str(p) for p in sieve.primerange(2, 10**n))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Apr 23 2021

Extensions

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 22 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091705 Number of primes less than 10^n having at least one digit 4.

Original entry on oeis.org

0, 3, 32, 326, 3231, 31953, 310456, 3000349, 28922364, 278508004, 2680391313, 25793058269, 248228108241, 2389543008906
Offset: 1

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Examples

			a(2) = 3 because of the 25 primes less than 10^2, 3 have at least one digit 4.

Crossrefs

a(n) + A091638(n) = A006880(n).
Cf. A091644, A091645, A091646, A091647, A091706, A091707, A091708, A091709, A091710.

Programs

  • Mathematica
    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], 4] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)

Extensions

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 21 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015

A091706 Number of primes less than 10^n having at least one digit 5.

Original entry on oeis.org

1, 3, 32, 324, 3282, 31949, 309669, 2995706, 28891689, 278270868, 2678674624, 25780340194, 248131741834, 2388797604792
Offset: 1

Views

Author

Enoch Haga, Jan 30 2004

Keywords

Examples

			a(1) = 1 because of the 4 primes less than 10^1, 1 has at least one digit 5.

Crossrefs

a(n) + A091639(n) = A006880(n).
Cf. A091644, A091645, A091646, A091647, A091705, A091707, A091708, A091709, A091710.

Programs

  • Mathematica
    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], 5] != {}, c++ ]]; Print[c]; p--, {n, 1, 8}] (* Robert G. Wilson v, Feb 02 2004 *)
Table[Count[Prime[Range[PrimePi[10^n]]],?(DigitCount[#,10,5]>0&)],{n,8}] (* _Harvey P. Dale, Dec 29 2012 *)

Extensions

Edited and extended by Robert G. Wilson v, Feb 02 2004
3 more terms from Ryan Propper, Aug 21 2005
a(13) from Robert Price, Nov 11 2013
a(14) from Giovanni Resta, Jul 21 2015
Showing 1-10 of 10 results.

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