A065587 -id:A065587 - cp's OEIS frontend

A037061 Smallest prime containing exactly n 4's.

2, 41, 443, 4441, 44449, 444443, 24444443, 424444441, 444444443, 4444444447, 44444444441, 444444444443, 14444444444449, 440444444444441, 2444444444444447, 44044444444444441, 424444444444444447, 4344444444444444449, 42444444444444444443, 44444444444444444447

Offset: 0

Patrick De Geest, Jan 04 1999

The last digit of n cannot be 4, therefore a(n) must have at least n+1 digits. It is probable that none among [10^n/9]*40 + {1,3,7,9} is prime in which case a(n) must have n+2 digits. We conjecture that for all n >= 0, a(n) equals [10^(n+1)/9]*40 + b with 1 <= b <= 9 and one of the (first) digits 4 replaced by a 0, 1, 2 or 3. - M. F. Hasler, Feb 22 2016

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

Cf. A065587, A037060, A034388, A036507-A036536.

Cf. A037053, A037055, A037057, A037059, A037063, A037065, A037067, A037069, A037071.

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, 4], {n, 1, 18}]

A037061(n)={my(p, t=10^(n+1)\9*40); forvec(v=[[-1, n], [-4, -1]], nextprime(p=t+10^(n-v[1])*v[2])-p<10 && return(nextprime(p)))} \\ M. F. Hasler, Feb 22 2016

a(n) = prime(A037060(n)). - Amiram Eldar, Jul 21 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003

More terms and a(0) = 2 from M. F. Hasler, Feb 22 2016

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

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

A037061 Smallest prime containing exactly n 4's.

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A068105 Smallest prime starting with n 5s.

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A065589 Smallest prime beginning with exactly n 6's.

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