A068104 -id:A068104 - cp's OEIS frontend

A065584 Smallest prime beginning with exactly n 1's.

2, 13, 11, 1117, 11113, 111119, 11111101, 11111117, 111111113, 11111111129, 11111111113, 1111111111139, 11111111111123, 1111111111111013, 1111111111111123, 11111111111111101, 11111111111111119

Offset: 0

Robert G. Wilson v, Nov 28 2001

Largest terms are only probable primes. Some terms of the sequence are also in A004022 (repunit primes) or A056710 (all digits same except last digit). All terms of A004022 are in the current sequence.

According to the Magma Calculator (at http://magma.maths.usyd.edu.au/calc/), all 201 terms in the table (and thus all 17 terms listed above) are, in fact, prime. - Jon E. Schoenfield, Aug 24 2009

M. F. Hasler, Table of n, a(n) for n = 0..200

Cf. A065821, A068103, A068104, A068105.

Cf. A004022 (repunit primes), A056710 (near-repdigit primes).

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

A099656 a(n) is the least prime following A002276(n) repdigits.

Original entry on oeis.org

2, 3, 23, 223, 2237, 22229, 222247, 2222239, 22222223, 222222227, 2222222243, 22222222223, 222222222301, 2222222222243, 22222222222229, 222222222222227, 2222222222222281, 22222222222222301, 222222222222222281

Offset: 0

Labos Elemer, Nov 17 2004

			n=2: 22 is followed by 23.

Robert Israel, Table of n, a(n) for n = 0..999

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283.

seq(nextprime(2*(10^i-1)/9), i=0..20); # Robert Israel, Aug 25 2017

Table[NextPrime[2*(10^n-1)/9], {n, 0, 35}]
Table[ NextPrime[2*(10^n - 1)/9], {n, 0, 18}] (* Robert G. Wilson v, Nov 20 2004 *)
Table[NextPrime[FromDigits[PadRight[{},n,2]]],{n,0,20}] (* Harvey P. Dale, Dec 15 2021 *)

A099668 a(n) is the largest prime before A002282(n) repdigits.

Original entry on oeis.org

7, 83, 887, 8887, 88883, 888887, 8888861, 88888883, 888888887, 8888888837, 88888888859, 888888888887, 8888888888857, 88888888888873, 888888888888883, 8888888888888753, 88888888888888801, 888888888888888859, 8888888888888888881, 88888888888888888879, 888888888888888888857

Offset: 1

Labos Elemer, Nov 17 2004

			n=2: 83 is before 88.

Cf. A002282, A003618, A003617, A007917, A096497, A096498, A068104, A002275-A002283, A099656-A099667.

Table[NextPrime[8*(10^n-1)/9, -1], {n, 1, 35}]
Table[NextPrime[FromDigits[PadRight[{},n,8]],-1],{n,20}] (* Harvey P. Dale, Jul 12 2014 *)

a(n) = A007917(A002282(n)). - Amiram Eldar, Jun 29 2025

A068103 Smallest prime starting with at least n 2s.

Original entry on oeis.org

2, 2, 223, 2221, 22229, 2222203, 22222223, 22222223, 222222227, 22222222223, 22222222223, 2222222222243, 22222222222201, 22222222222229, 222222222222227, 222222222222222043, 222222222222222221

Offset: 0

Amarnath Murthy, Feb 20 2002

M. F. Hasler, Table of n, a(n) for n = 0..500

Cf. A065585, A065584, A068104, A068105.

A068103(n)={n=10^n\9*2;n>2&for(d=1,9e9,n*=10;for(t=1,10^d-1,ispseudoprime(n+t)&return(n+t)));2} \\ - M. F. Hasler, Oct 17 2012

More terms from Sascha Kurz, Mar 19 2002

Corrected by Don Reble, Jan 17 2007

A068105 Smallest prime starting with n 5s.

Original entry on oeis.org

2, 5, 557, 5557, 555521, 555557, 55555517, 55555553, 5555555501, 5555555557, 555555555551, 555555555551, 5555555555551, 555555555555529, 555555555555557, 55555555555555519, 555555555555555559, 555555555555555559, 55555555555555555567, 5555555555555555555087

Offset: 0

Amarnath Murthy, Feb 20 2002

Michael S. Branicky, Table of n, a(n) for n = 0..1000

Cf. A065584, A068103, A068104, A065587, A065588.

from sympy import isprime
def a(n):
  if n < 2: return list([2, 5])[n]
  n5s, i, pow10 = int('5'*n), 1, 1
  while True:
    i = 1
    while i < pow10:
      t = n5s * pow10 + i
      if isprime(t): return t
      i += 2
    pow10 *= 10
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 05 2021

a(n) <= A065588(n). - Michael S. Branicky, Feb 05 2021

Edited and extended by Robert G. Wilson v, Feb 21 2002

Corrected by Don Reble, Jan 17 2007

A099667 a(n) is the largest prime before A002281(n); repdigits repeating 7.

Original entry on oeis.org

5, 73, 773, 7759, 77773, 777769, 7777769, 77777761, 777777773, 7777777741, 77777777767, 777777777773, 7777777777771, 77777777777753, 777777777777773, 7777777777777753, 77777777777777747, 777777777777777743

Offset: 1

Labos Elemer, Nov 17 2004

			n=2: 73 is before 77.

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[7 (10^n - 1)/9, -1], {n, 35}]
(* Second program: *)
Rest[NextPrime[#,-1]&/@LinearRecurrence[{11,-10},{0,7},25]] (* Harvey P. Dale, May 24 2015 *)

Name changed by David A. Corneth, Sep 01 2017

A123568 Prime numbers of the form (10^n - 7)/3.

Original entry on oeis.org

31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331

Offset: 1

Artur Jasinski, Nov 12 2006

nonn

The number of initial 3s is n - 1.

Note that each n from 2 to 8 gives primes, but after that the n that correspond to primes are progressively further apart. Singh (1997) gives this as an example of why mathematicians don't trust a preponderance of evidence as proof: in the 17th century, when factoring numbers with as few as eight digits wasn't as easy as it is today, the pattern suggested that all numbers of this form are prime. - Alonso del Arte, Nov 11 2012

			a(7) = 33333331 because that is the seventh number of the specified form to be prime.
333333331 is not in the sequence because it is composite, being the product of 17 and 19607843.

Simon Singh, Fermat's Enigma. New York: Walker & Company (1997) p. 159.

Alois P. Heinz, Table of n, a(n) for n = 1..17

Cf. A055557, A051200, A033175, A065586, A068104.

Do[If[PrimeQ[(10^n - 7)/3], Print[(10^n - 7)/3]], {n, 1, 100}] (* Jasinski *)
Select[(10^Range[50] - 7)/3, PrimeQ[#] &] (* Alonso del Arte, Nov 11 2012 *)
Select[Table[FromDigits[PadLeft[{1},n,3]],{n,50}],PrimeQ] (* Harvey P. Dale, Dec 05 2018 *)

select(ispseudoprime, vector(20, n, (10^n-7)/3)) \\ Charles R Greathouse IV, Nov 12 2012

A099658 a(n) is the smallest prime greater than 4(10^n - 1)/9.

Original entry on oeis.org

2, 5, 47, 449, 4447, 44449, 444449, 4444469, 44444453, 444444457, 4444444447, 44444444497, 444444444461, 4444444444493, 44444444444459, 444444444444461, 4444444444444463, 44444444444444461, 444444444444444469, 4444444444444444537, 44444444444444444447

Offset: 0

Labos Elemer, Nov 17 2004

a(n) = smallest prime > A002278(n).

			n=4: 44 is followed by 47.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[4*(10^n-1)/9], {n, 0, 35}]

Checked by N. J. A. Sloane, Jan 27 2007

Mathematica program edited by Harvey P. Dale, Jul 16 2024

A099659 a(n) is the least prime following A002279(n) repdigits.

Original entry on oeis.org

7, 59, 557, 5557, 55579, 555557, 5555567, 55555559, 555555587, 5555555557, 55555555619, 555555555559, 5555555555593, 55555555555573, 555555555555557, 5555555555555573, 55555555555555639, 555555555555555559, 5555555555555555621, 55555555555555555567, 555555555555555555619

Offset: 1

Labos Elemer, Nov 17 2004

			n=2: 55 is followed by 59.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[5*(10^n-1)/9], {n, 1, 35}]

Offset corrected by Georg Fischer, Mar 12 2024

A099660 a(n) is the least prime following A002280[n] repdigits.

Original entry on oeis.org

2, 7, 67, 673, 6673, 66683, 666667, 6666679, 66666667, 666666667, 6666666757, 66666666667, 666666666671, 6666666666683, 66666666666737, 666666666666719, 6666666666666719, 66666666666666713, 666666666666666773

Offset: 0

Labos Elemer, Nov 17 2004

			n=6: 66 is followed by 67.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[6*(10^n-1)/9], {n, 0, 35}]
NextPrime/@Table[FromDigits[PadRight[{},n,6]],{n,0,20}] (* Harvey P. Dale, Feb 25 2015 *)

First Mathematica program corrected by Harvey P. Dale, Feb 25 2015

cp's OEIS Frontend

A065584 Smallest prime beginning with exactly n 1's.

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A099656 a(n) is the least prime following A002276(n) repdigits.

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A099668 a(n) is the largest prime before A002282(n) repdigits.

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A068103 Smallest prime starting with at least n 2s.

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A068105 Smallest prime starting with n 5s.

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A099667 a(n) is the largest prime before A002281(n); repdigits repeating 7.

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A123568 Prime numbers of the form (10^n - 7)/3.

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Mathematica

PARI

A099658 a(n) is the smallest prime greater than 4(10^n - 1)/9.

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Mathematica

Extensions

A099659 a(n) is the least prime following A002279(n) repdigits.