A065584 -id:A065584 - cp's OEIS frontend

A037055 Smallest prime containing exactly n 1's.

2, 13, 11, 1117, 10111, 101111, 1111151, 11110111, 101111111, 1111111121, 11111111113, 101111111111, 1111111118111, 11111111111411, 111111111116111, 1111111111111181, 11111111101111111, 101111111111111111, 1111111111111111171, 1111111111111111111, 111111111111111119111

Offset: 0

Patrick De Geest, Jan 04 1999

For n > 1, A037055 is conjectured to be identical to A084673. - Robert G. Wilson v, Jul 04 2003

a(n) = A002275(n) for n in A004023. For all other n < 900, a(n) has n+1 digits. - Robert Israel, Feb 21 2016

Robert Israel, Table of n, a(n) for n = 0..900

Cf. A084673, A065584, A065821, A037054, A034388, A036507-A036536.

Cf. A037053, A037057, A037059, A037061, A037063, A037065, A037067, A037069, A037071.

f:= proc(n) local m,d,r,x;
   r:= (10^n-1)/9;
   if isprime(r) then return r fi;
   r:= (10^(n+1)-1)/9;
   for m from n-1 to 1 by -1 do
     x:= r - 10^m;
     if isprime(x) then return x fi;
   od;
   for m from 0 to n do
     for d from 1 to 8 do
        x:= r + d*10^m;
        if isprime(x) then return x fi;
     od
   od;
   error("Needs more than n+1 digits")
end proc:
map(f, [$0..100]); # Robert Israel, Feb 21 2016

f[n_, b_] := Block[{k = 10^(n + 1), p = Permutations[ Join[ Table[b, {i, 1, n}], {x}]], c = Complement[Table[j, {j, 0, 9}], {b}], q = {}}, Do[q = Append[q, Replace[p, x -> c[[i]], 2]], {i, 1, 9}]; r = Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]; If[r ? Infinity, r, p = Permutations[ Join[ Table[ b, {i, 1, n}], {x, y}]]; q = {}; Do[q = Append[q, Replace[p, {x -> c[[i]], y -> c[[j]]}, 2]], {i, 1, 9}, {j, 1, 9}]; Min[ Select[ FromDigits /@ Flatten[q, 1], PrimeQ[ # ] & ]]]]; Table[ f[n, 1], {n, 1, 18}]
Join[{2, 13}, Table[Sort[Flatten[Table[Select[FromDigits/@Permutations[Join[{n}, PadRight[{}, i, 1]]], PrimeQ], {n, 0, 9}]]][[1]], {i, 2, 20}]] (* Vincenzo Librandi, May 11 2017 *)

A037055(n)={my(p,t=10^(n+1)\9); forstep(k=n+1,1,-1, ispseudoprime(p=t-10^k) && return(p)); forvec(v=[[0, n], [1, 8]], ispseudoprime(p=t+10^v[1]*v[2]) && return(p))} \\ M. F. Hasler, Feb 22 2016

a(n) = the smallest prime in { R-10^n, R-10^(n-1), ..., R-10; R+a*10^b, a=1, ..., 8, b=0, 1, 2, ..., n }, where R = (10^(n+1)-1)/9 is the (n+1)-digit repunit. - M. F. Hasler, Feb 25 2016

a(n) = prime(A037054(n)). - Amiram Eldar, Jul 21 2025

More terms from Sascha Kurz, Feb 10 2003
Edited by Robert G. Wilson v, Jul 04 2003
a(0) = 2 inserted by Robert Israel, Feb 21 2016

A068104 Smallest prime starting with n 3s.

Original entry on oeis.org

2, 3, 331, 3331, 33331, 333331, 3333331, 33333331, 3333333319, 33333333329, 333333333323, 3333333333301, 33333333333319, 333333333333307, 3333333333333301, 33333333333333323, 333333333333333331

Offset: 0

Amarnath Murthy, Feb 20 2002

Cf. A065584, A068103, A068105.

Join[{2,3},Table[SelectFirst[Join[10FromDigits[PadRight[{},k,3]]+{1,7,9},Flatten[Table[100 FromDigits[PadRight[{},k,3]]+10n+{1,3,7,9},{n,0,9}]],Flatten[Table[1000 FromDigits[PadRight[{},k,3]]+100n+{1,3,7,9},{n,0,99}]]],PrimeQ],{k,2,20}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 01 2019 *)

More terms from Sascha Kurz, Mar 19 2002

A065592 Smallest prime beginning with exactly n 9's.

Original entry on oeis.org

2, 97, 991, 99901, 99991, 9999901, 9999991, 999999929, 9999999929, 99999999907, 999999999937, 9999999999971, 99999999999923, 999999999999947, 9999999999999917, 99999999999999919, 99999999999999997, 9999999999999999919, 99999999999999999931

Offset: 0

Robert G. Wilson v, Nov 28 2001

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

Cf. A037071, A065582, A065584 - A065591.

fp[n_]:=Select[Join[10*n+{1,7},100*n+Range[1,99,2]],PrimeQ,1]; With[{ns=Table[FromDigits[PadRight[{},n,9]],{n,20}]}, Join[{2}, Flatten[fp/@ns]]] (* Harvey P. Dale, May 12 2012 *)

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

A065821 a(n) is the smallest prime ending in exactly n 1's.

Original entry on oeis.org

31, 11, 2111, 101111, 311111, 29111111, 61111111, 1711111111, 14111111111, 31111111111, 311111111111, 2111111111111, 31111111111111, 3511111111111111, 5111111111111111, 101111111111111111, 3511111111111111111, 2111111111111111111, 1111111111111111111, 911111111111111111111

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Nov 23 2001

			a(4) = 101111 because 1111=11*101, 21111=3*31*227, 31111=53*587, 41111=7^2*829, 51111=3^4*631, 61111=23*2657, 71111=17*47*89, 81111=3*19*1423, 91111=179*509 so 101111 is the first prime ending in four 1's.

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A037055, A065584, A065580, A065581, A065582.

pe[n_]:=Module[{k=0,len=IntegerLength[n]},While[Mod[k,10]==1||(!PrimeQ[ k*10^len+n]),k++];k*10^len+n]; pe/@Table[(10^n-1)/9,{n,20}] (* Harvey P. Dale, Dec 31 2013 *)_

a(n)={ my(f=10^n, b=(f-1)/9, k=0); while (!isprime(b + k*f), k+=1+(k%10==0)); b + k*f } \\ Harry J. Smith, Nov 01 2009

Edited and extended by Robert G. Wilson v, Jul 04 2003

A065585 Smallest prime beginning with exactly n 2's.

Original entry on oeis.org

3, 2, 223, 2221, 22229, 2222203, 22222253, 22222223, 222222227, 22222222273, 22222222223, 2222222222243, 22222222222201, 22222222222229, 222222222222227, 222222222222222043, 222222222222222281, 222222222222222221, 22222222222222222253, 222222222222222222277

Offset: 0

Robert G. Wilson v, Nov 28 2001

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

Cf. A037057, A065584 - A065592.

A068103 is a lower bound, but most often equality holds. - M. F. Hasler, Oct 17 2012

Do[a = Table[2, {n}]; k = 0; While[b = FromDigits[ Join[a, IntegerDigits[k] ]]; First[ IntegerDigits[k]] == 2 || !PrimeQ[b], k++ ]; Print[b], {n, 1, 17} ]

A065585(n)={n=10^n\9*2; n>2&for(d=1, 9e9, n*=10; for(t=1, 10^d-1, t\10^(d-1)==2 & t+= 10^(d-1)+(t>2); ispseudoprime(n+t) & return(n+t))); 2+!n} \\ M. F. Hasler, Oct 17 2012

from sympy import isprime
def a(n):
  if n < 2: return list([3, 2])[n]
  n2s, i, pow10, end_digits = int('2'*n), 1, 1, 0
  while True:
    i = 1
    while i < pow10:
      istr = str(i)
      if istr[0] == '2' and len(istr) == end_digits:
        i += pow10 // 10
      else:
        t = n2s * pow10 + i
        if isprime(t): return t
        i += 2
    pow10 *= 10; end_digits += 1
print([a(n) for n in range(20)]) # Michael S. Branicky, Mar 02 2021

Corrected by Don Reble, Jan 17 2007

A065586 Smallest prime beginning with exactly n 3's.

Original entry on oeis.org

2, 3, 331, 3331, 33331, 333331, 3333331, 33333331, 3333333319, 33333333329, 333333333323, 3333333333301, 33333333333319, 333333333333307, 3333333333333301, 33333333333333323, 333333333333333391, 333333333333333331, 33333333333333333359, 3333333333333333333041

Offset: 0

Robert G. Wilson v, Nov 28 2001

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

Cf. A037059, A065580, A065584 - A065592.

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

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

A065587 Smallest prime beginning with exactly n 4's.

Original entry on oeis.org

2, 41, 443, 4441, 44449, 444443, 44444453, 444444421, 444444443, 4444444447, 44444444441, 444444444443, 44444444444459, 444444444444421, 4444444444444423, 44444444444444411, 444444444444444413, 4444444444444444409, 44444444444444444479, 44444444444444444447

Offset: 0

Robert G. Wilson v, Nov 28 2001

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

Cf. A037061, A065584 - A065592.

Offset corrected by Sean A. Irvine, Sep 06 2023

A065588 Smallest prime beginning with exactly n 5's.

Original entry on oeis.org

2, 5, 557, 5557, 555521, 555557, 55555517, 55555553, 5555555501, 5555555557, 5555555555057, 555555555551, 5555555555551, 555555555555529, 555555555555557, 55555555555555519, 5555555555555555021, 555555555555555559, 55555555555555555567, 5555555555555555555087

Offset: 0

Robert G. Wilson v, Nov 28 2001

Michael S. Branicky, Table of n, a(n) for n = 0..997 (terms 0..200 from M. F. Hasler)

Cf. A037063, A065584 - A065592. Different from A068105.

from sympy import isprime
def a(n):
  if n < 2: return list([2, 5])[n]
  n5s, i, pow10, end_digits = int('5'*n), 1, 1, 0
  while True:
    i = 1
    while i < pow10:
      istr = str(i)
      if istr[0] == '5' and len(istr) == end_digits:
        i += 2 if pow10 <= 10 else pow10 // 10
      else:
        t = n5s * pow10 + i
        if isprime(t): return t
        i += 2
    pow10 *= 10; end_digits += 1
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 05 2021, corrected Mar 03 2021

Corrected by N. J. A. Sloane, Jan 11 2007 and by Don Reble, Jan 17 2007

Offset corrected by Michael S. Branicky, Feb 05 2021

cp's OEIS Frontend

A037055 Smallest prime containing exactly n 1's.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A068104 Smallest prime starting with n 3s.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A065592 Smallest prime beginning with exactly n 9's.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A065821 a(n) is the smallest prime ending in exactly n 1's.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A065585 Smallest prime beginning with exactly n 2's.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A065586 Smallest prime beginning with exactly n 3's.

Views

Author

Keywords

Links

Crossrefs

Extensions

A068103 Smallest prime starting with at least n 2s.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A068105 Smallest prime starting with n 5s.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Formula

Extensions

A065587 Smallest prime beginning with exactly n 4's.