A065592 -id:A065592 - cp's OEIS frontend

A037071 Smallest prime containing exactly n 9's.

2, 19, 199, 1999, 49999, 199999, 2999999, 19999999, 799999999, 9199999999, 59999999999, 959999999999, 9919999999999, 59999999999999, 499999999999999, 9299999999999999, 99919999999999999, 994999999999999999, 9991999999999999999, 29999999999999999999, 989999999999999999999

Offset: 0

Patrick De Geest, Jan 04 1999

We conjecture that for all n >= 0, a(n) equals [10^(n+1)/9]*9 with one of the (first) digits 9 replaced by a digit among {1, 2, 4, 5, 7, 8}. - M. F. Hasler, Feb 22 2016

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

Cf. A065592, A065582, A037070, A034388, A036507-A036536.

Cf. A037053, A037055, A037057, A037059, A037061, A037063, A037065, A037067, A037069.

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, 9], {n, 1, 20}]

A037071(n)={my(t=10^(n+1)\9*9); forvec(v=[[-1, n], [-8, -1]], ispseudoprime(p=t+10^(n-v[1])*v[2]) && return(p));error} \\ M. F. Hasler, Feb 22 2016

a(n) = prime(A037070(n)). - Amiram Eldar, Jul 21 2025

More terms from Vladeta Jovovic, Jan 10 2002

a(0) = 2 prepended by M. F. Hasler, Feb 22 2016

A065582 Smallest prime ending in exactly n 9's.

Original entry on oeis.org

19, 199, 1999, 49999, 199999, 2999999, 19999999, 799999999, 10999999999, 59999999999, 1099999999999, 34999999999999, 59999999999999, 499999999999999, 14999999999999999, 139999999999999999, 1099999999999999999, 20999999999999999999, 29999999999999999999, 2099999999999999999999

Offset: 1

Robert G. Wilson v, Nov 28 2001

From Jeppe Stig Nielsen, Jul 30 2022: (Start)
Can decrease, for example a(25) < a(24). So not the same as Smallest prime ending in n or more 9s.
a(n) can contain other 9s as well, for example a(46), a(118), a(156). (End)

Hugo Pfoertner, Table of n, a(n) for n = 1..500 (first 100 terms from Harry J. Smith).

Cf. A037071, A065592, A065821, A065580, A065581.

Do[a = Table[9, {n} ]; k = 0; While[ b = FromDigits[ Join[ IntegerDigits[k], a]]; Mod[k, 10] == 9 || !PrimeQ[b], k++ ]; Print[b], {n, 1, 17} ]

a(n)={ my(t=10^n, b=t-1, d=0); while(!isprime(b + t*d), d++; if(d%10==9, d++)); b + t*d } \\ Harry J. Smith, Oct 23 2009

from sympy import isprime
def a(n):
    pow, end, i = 10**n, 10**n-1, 1
    while i%10 == 9 or not isprime(i*pow + end): i += 1
    return i*pow + end
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Jul 30 2022

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

A065480 Decimal expansion of Product_{p prime} (1 - 1/(p^2+p-1)).

Original entry on oeis.org

6, 6, 9, 5, 8, 0, 2, 9, 0, 5, 3, 9, 0, 6, 2, 3, 6, 7, 6, 3, 5, 0, 2, 5, 6, 9, 5, 6, 1, 2, 4, 3, 4, 2, 2, 7, 2, 1, 7, 3, 3, 9, 8, 2, 5, 4, 1, 6, 2, 3, 3, 0, 2, 5, 6, 2, 4, 6, 5, 4, 6, 2, 6, 3, 3, 0, 9, 8, 3, 6, 6, 1, 9, 9, 5, 4, 7, 2, 4, 5, 7, 1, 4, 5, 7, 5, 6, 6, 2, 6, 0, 3, 8, 6, 9, 6, 3, 8

Offset: 0

N. J. A. Sloane, Nov 19 2001

The probability that two numbers are coprime given that they are both coprime to a randomly chosen third number. - Luke Palmer, Apr 27 2019

			0.6695802905390623676350256956124342...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A065592, A078081.

Cf. A013661, A065463, A065473.

digits = 98; Exp[NSum[(1/2)*(-2 + (-2)^n - ((1/2)*(-1 - Sqrt[5]))^n*(-1 + Sqrt[5]) + ((1/2)*(-1 + Sqrt[5]))^n*(1 + Sqrt[5]))*PrimeZetaP[n - 1]/(n - 1), {n, 3, Infinity}, WorkingPrecision -> 4 digits, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p^2+p-1)) \\ Amiram Eldar, Mar 14 2021

Equals A065473*zeta(2)/A065463. - Luke Palmer, Apr 27 2019

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

A065590 Smallest prime beginning with exactly n 7's.

Original entry on oeis.org

2, 7, 773, 77711, 77773, 7777709, 77777719, 777777701, 777777773, 77777777717, 777777777713, 777777777773, 7777777777771, 777777777777719, 777777777777773, 77777777777777711, 777777777777777737, 7777777777777777793, 77777777777777777729, 77777777777777777771

Offset: 0

Robert G. Wilson v, Nov 28 2001

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

Cf. A037067, A065581, A065584 - A065592.

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

A065591 Smallest prime beginning with exactly n 8's.

Original entry on oeis.org

2, 83, 881, 8887, 88883, 888887, 88888817, 88888883, 888888883, 88888888801, 888888888859, 888888888887, 88888888888873, 88888888888889, 888888888888883, 88888888888888801, 88888888888888889, 8888888888888888813, 8888888888888888881, 888888888888888888857

Offset: 0

Robert G. Wilson v, Nov 28 2001

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

Cf. A037069, A065584 - A065592.

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

A065589 Smallest prime beginning with exactly n 6's.

Original entry on oeis.org

2, 61, 661, 6661, 666607, 666667, 66666629, 66666667, 666666667, 6666666661, 66666666667, 6666666666629, 66666666666629, 666666666666631, 66666666666666047, 66666666666666601, 6666666666666666059, 666666666666666661, 66666666666666666601, 66666666666666666667

Offset: 0

Robert G. Wilson v, Nov 28 2001

David A. Corneth, Table of n, a(n) for n = 0..700 (first 201 terms from M. F. Hasler)

Cf. A037065, A065584, A065585, A065586, A065587, A065588, (this sequence), A065590, A065591, A065592.

Do[a = Table[6, {n}]; k = 0; While[b = FromDigits[ Join[a, IntegerDigits[k] ]]; First[ IntegerDigits[k]] == 6 || !PrimeQ[b], k++ ]; Print[b], {n, 1, 17} ]

a(n) = {if(n==0, return(2)); my(cs = 60*(10^n\9), pow10 = 10); for(i = 1, oo, np = cs; d = 0; while(d < pow10, np = nextprime(np + 1); d = np - cs; if(d < pow10 && digits(d)[1] != 6 || 10*d < pow10, return(np))); cs*=10; pow10*=10)} \\ David A. Corneth, Sep 06 2023

Corrected by Don Reble, Jan 17 2007

Offset corrected by Sean A. Irvine, Sep 06 2023

cp's OEIS Frontend

A037071 Smallest prime containing exactly n 9's.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A065582 Smallest prime ending in exactly n 9's.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

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

A065480 Decimal expansion of Product_{p prime} (1 - 1/(p^2+p-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065587 Smallest prime beginning with exactly n 4's.

Views

Author

Keywords

Links

Crossrefs

Extensions

A065588 Smallest prime beginning with exactly n 5's.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Extensions

A065590 Smallest prime beginning with exactly n 7's.

Views

Author

Keywords

Links

Crossrefs

Extensions

A065591 Smallest prime beginning with exactly n 8's.

Views

Author