A266147 -id:A266147 - cp's OEIS frontend

A266148 Number of n-digit primes in which n-1 of the digits are 9's.

4, 6, 7, 7, 8, 10, 7, 13, 8, 8, 11, 13, 8, 11, 13, 14, 10, 9, 7, 11, 9, 13, 10, 19, 5, 10, 14, 7, 10, 9, 9, 15, 13, 8, 7, 9, 10, 11, 10, 13, 5, 12, 15, 7, 12, 7, 12, 11, 13, 11, 8, 13, 13, 13, 12, 12, 9, 9, 15, 14, 9, 8, 13, 11, 15, 17, 10, 8, 11, 10, 6, 16, 8, 8, 8, 15, 9, 11, 14, 7, 10, 11, 16, 17, 11, 10, 12, 16, 8, 15, 7, 11, 11, 10, 7, 12, 6, 10, 8, 9

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

The other digit cannot be 0, 3, 6, or 9, or else the number would not be prime. - N. J. A. Sloane, May 20 2016

			a(3) = 7 since 199, 499, 599, 919, 929, 991 and 997 are all the three-digit primes containing two 9's.

Dana Jacobsen, Table of n, a(n) for n = 1..2500 (first 1215 terms from Michael De Vlieger and Robert G. Wilson v)

Cf. A265733, A266141, A266142, A266143, A266144, A266145, A266146, A266147, A266149, A095714, A089675.

f9[n_] := Block[{cnt = k = 0, r = 9 (10^n - 1)/9, s = Range[0, 9] - 9}, While[k < n, cnt += Length@ Select[r + 10^k * s, PrimeQ@ # && IntegerLength@ # > k &]; k++]; cnt]; Array[f9, 100]

use ntheory ":all"; sub a266148 { my $n = shift; vecsum( map { my $k=$; scalar grep { is_prime("9" x $k . $ . "9" x ($n-$k-1)) } 0+($k>0) .. 8 } 0 .. $n-1 ); } # Dana Jacobsen, Jan 01 2016

from sympy import isprime
def A266148(n):
    return sum(1 for d in range(-9,1) for i in range(n) if isprime(10**n-1+d*10**i)) # Chai Wah Wu, Dec 31 2015

A266142 Number of n-digit primes in which n-1 of the digits are 3's.

Original entry on oeis.org

4, 8, 9, 12, 7, 14, 13, 11, 8, 7, 9, 8, 3, 10, 11, 14, 9, 12, 6, 11, 11, 11, 9, 10, 9, 10, 22, 10, 10, 12, 7, 14, 14, 15, 7, 16, 11, 7, 14, 10, 13, 13, 8, 10, 11, 12, 6, 12, 10, 10, 10, 11, 5, 14, 8, 8, 5, 14, 6, 18, 13, 9, 13, 10, 4, 14, 12, 6, 11, 13, 12, 20, 11, 9, 13, 6, 12, 22, 13, 10, 10, 12, 5, 20, 11, 10, 11, 10, 11, 12, 11, 13, 12, 18, 7, 20, 15, 6, 8, 8, 8, 15, 12, 10, 14

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

			a(2) = 8 since 13, 23, 31, 37, 43, 53, 73 and 83 are all primes.
a(3) = 9 since 233, 313, 331, 337, 353, 373, 383, 433 and 733 are all primes.

Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1215

Cf. A265733, A266141, A266143, A266144, A266145, A266146, A266147, A266148, A266149, A055557, A099411.

f3[n_] := Block[{cnt = k = 0, r = 3 (10^n - 1)/9, s = Range[0, 9] - 3}, While[k < n, cnt += Length@ Select[r + 10^k*s, PrimeQ@ # && IntegerLength@ # > k &]; k++]; cnt]; Array[f3, 105]

a(n)={sum(i=0 ,n-1, sum(d=i==n-1, 9, isprime((10^n-1)/3 + (d-3)*10^i)))} \\ Andrew Howroyd, Feb 28 2018

from _future_ import division
from sympy import isprime
def A266142(n):
    return 4*n if (n==1 or n==2) else sum(1 for d in range(-3,7) for i in range(n) if isprime((10**n-1)//3+d*10**i)) # Chai Wah Wu, Dec 27 2015

a(2) corrected by Chai Wah Wu, Dec 27 2015

a(2) in b-file corrected as above by Andrew Howroyd, Feb 28 2018

A266146 Number of n-digit primes in which n-1 of the digits are 7's.

Original entry on oeis.org

4, 8, 10, 9, 12, 11, 8, 4, 9, 9, 10, 14, 14, 11, 16, 7, 10, 17, 7, 10, 9, 12, 9, 13, 11, 10, 14, 5, 3, 22, 6, 13, 13, 10, 8, 16, 8, 6, 16, 8, 13, 14, 8, 7, 8, 13, 9, 11, 13, 9, 14, 8, 4, 23, 13, 11, 8, 8, 8, 12, 13, 13, 11, 11, 10, 23, 11, 8, 8, 3, 6, 16, 12, 13, 12, 12, 8, 11, 8, 11, 14, 13, 7, 15, 12, 17, 11, 7, 9, 21, 6, 6, 11, 12, 6, 14, 14, 12, 13, 12, 11, 17, 10, 17, 18

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

			a(2) = 8 from 17, 37, 47, 67, 71, 73, 79, 97. - _N. J. A. Sloane_, Dec 27 2015
a(3) = 10 since 277, 577, 677, 727, 757, 773, 787, 797, 877, and 977 are primes.

Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1215

Cf. A265733, A266141, A266142, A266143, A266144, A266145, A266147, A266148, A266149, A099419, A099420, A098089.

f7[n_] := Block[{cnt = k = 0, r = 7 (10^n - 1)/9, s = Range[0, 9] - 7}, While[k < n, cnt += Length@ Select[r + 10^k*s, PrimeQ@ # && IntegerLength@ # > k &]; k++]; cnt]; Array[f7, 100]

a(n)={sum(i=0, n-1, sum(d=i==n-1, 9, isprime((10^n-1)/9*7 + (d-7)*10^i)))} \\ Andrew Howroyd, Feb 28 2018

from _future_ import division
from sympy import isprime
def A266146(n):
     return 4*n if (n==1 or n==2) else sum(1 for d in range(-7,3) for i in range(n) if isprime(7*(10**n-1)//9+d*10**i)) # Chai Wah Wu, Dec 27 2015

a(2) corrected by Chai Wah Wu, Dec 27 2015

a(2) corrected in b-file as above by Andrew Howroyd, Feb 28 2018

A266141 Number of n-digit primes in which n-1 of the digits are 2's.

Original entry on oeis.org

4, 2, 3, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

The leading digits must be 2's and only the trailing digit can vary.
For n large a(n) is usually zero.
a(n) <= 4.  If n > 1 and not a multiple of 3, then a(n) <= 2.  It appears that a(n) <= 1 for n > 3. - Chai Wah Wu, Dec 26 2015

			a(3) = 3 since 223, 227 and 229 are all primes.

Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 1500 terms from Michael De Vlieger and Robert G. Wilson v)

Cf. A265733, A266142, A266143, A266144, A266145, A266146, A266147, A266148, A266149, A084832, A096506, A099409, A099410.

d = 2; Array[Length@ Select[d (10^# - 1)/9 + (Range[0, 9] - d), PrimeQ] &, 100]

use ntheory ":all"; sub a266141 { my $n=shift; return 4 if $n==1; 0+scalar(grep{is_prime("2"x($n-1).$)} 1,3,7,9); } say a266141($) for 1..20; # Dana Jacobsen, Dec 27 2015

from sympy import isprime
def A266141(n):
    return 4 if n==1 else sum(1 for d in '1379' if isprime(int('2'*(n-1)+d))) # Chai Wah Wu, Dec 26 2015

A266143 Number of n-digit primes in which n-1 of the digits are 4's.

Original entry on oeis.org

4, 3, 2, 2, 1, 2, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

The leading digits must be 4's and only the trailing digit can vary.

For n large a(n) is usually zero.

			a(3) = 2 since 443 and 449 are primes.
a(4) = 2 since 4441 and 4447 are primes.

Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1500

Cf. A265733, A266141, A266142, A266144, A266145, A266146, A266147, A266148, A266149, A099412, A096845, A099413, A099414.

d = 4; Array[Length@ Select[d (10^# - 1)/9 + (Range[0, 9] - d), PrimeQ] &, 100]

from _future_ import division
from sympy import isprime
def A266143(n):
    return 4 if n==1 else sum(1 for d in [-3,-1,3,5] if isprime(4*(10**n-1)//9+d)) # Chai Wah Wu, Dec 27 2015

A266144 Number of n-digit primes in which n-1 of the digits are 5's.

Original entry on oeis.org

4, 2, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

The leading digits must be 5's and only the trailing digit can vary.

For n large a(n) is usually zero.

			a(2) = 2 since 53 and 59 are primes.
a(3) = 1 since 557 is the only prime.

Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1500

Cf. A265733, A266141, A266142, A266143, A266145, A266146, A266147, A266148, A266149, A099415, A099416, A099417, A099418.

d = 5; Array[Length@ Select[d (10^# - 1)/9 + (Range[0, 9] - d), PrimeQ] &, 100]

from _future_ import division
from sympy import isprime
def A266144(n):
    return 4 if n==1 else sum(1 for d in [-4,-2,2,4] if isprime(5*(10**n-1)//9+d)) # Chai Wah Wu, Dec 27 2015

A266145 Number of n-digit primes in which n-1 of the digits are 6's.

Original entry on oeis.org

4, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

The leading digits must be 6's and only the trailing digit can vary.

For n large a(n) is usually zero.

			a(2) = 2 since 61 and 67 are prime.
a(3) = 1 since 661 is the only prime.

Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1500

Cf. A265733, A266141, A266142, A266143, A266144, A266146, A266147, A266148, A266149, A098088, A096507.

d = 6; Array[Length@ Select[d (10^# - 1)/9 + (Range[0, 9] - d), PrimeQ] &, 100]
Join[{4},Table[Count[Table[10FromDigits[PadRight[{},k,6]]+n,{n,{1,3,7,9}}], ?PrimeQ],{k,110}]] (* _Harvey P. Dale, Dec 23 2017 *)

from _future_ import division
from sympy import isprime
def A266145(n):
    return 4 if n==1 else sum(1 for d in [-5,-3,1,3] if isprime(2*(10**n-1)//3+d)) # Chai Wah Wu, Dec 27 2015

A266149 Number of n-digit primes that consist of at least n-1 copies of some decimal digit.

Original entry on oeis.org

4, 21, 46, 43, 40, 53, 35, 49, 40, 38, 44, 52, 35, 45, 49, 42, 38, 57, 28, 45, 38, 47, 38, 52, 33, 45, 56, 38, 36, 65, 29, 56, 48, 40, 38, 58, 37, 33, 57, 40, 37, 61, 41, 39, 37, 44, 36, 55, 47, 43, 47, 43, 35, 62, 43, 46, 29, 35, 37, 56, 39, 41, 46, 48, 39, 74, 45, 34, 34, 35, 34, 67, 39, 45, 43

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

The first n at which a(n)=k for k=1...80, or 0 if no such k exists with n < 701: 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 433, 141, 181, 847, 19, 31, 253, 357, 137, 25, 68, 7, 29, 37, 10, 44, 5, 43, 16, 4, 11, 14, 3, 22, 33, 8, 139, 82, 12, 6, 102, 48, 27, 18, 36, 270, 198, 42, 54, 498, 90, 30, 738, 72, 222, 192, 852, 84, 342, 0, 66, 0, 816, 264, 0, 288, 0.

			a(1) = 4 since 2, 3, 5 and 7 are primes,
a(2) = 21 since 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97 are primes,
a(3) = 46 since 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811, 877, 881, 883, 887, 911, 919, 929, 977, 991, 997 are all primes,
a(4) = 43 since 1117, 1151, 1171, 1181, 1511, 1777, 1811, 1999, 2111, 2221, 2333, 2777, 2999, 3313, 3323, 3331, 3343, 3373, 3433, 3533, 3733, 3833, 4111, 4441, 4447, 4999, 5333, 5557, 6661, 7177, 7333, 7477, 7577, 7717, 7727, 7757, 7877, 8111, 8887, 8999, 9199, 9929 and 9949 are primes; etc.

Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1215

Cf. A265733, A266141, A266142, A266143, A266144, A266145, A266146, A266147, A266148.

Length /@ Array[Function[n, Select[Union[Flatten[Function[k, Select[FromDigits /@ Flatten[Permutations[Flatten@ {Table[k, {n - 1}], #}] & /@ Range[0, 9], 1], PrimeQ]] /@ Range[1, 9]]], Function[m, IntegerLength@ m == n]]], 100] (* Michael De Vlieger, Jan 01 2016 *)

from sympy import isprime
def a(n):
  if n == 1: return 4
  okset = set()
  for digit1 in "24568":
    for digit2 in "1379":
      t = int(digit1*(n-1) + digit2)
      if isprime(t): okset.add(t)
  for digit1 in "1379":
    for digit2 in "0123456789":
      if ((n-1)*int(digit1) + int(digit2))%3 == 0: continue
      for j in range(n):
        mc = digit1*j + digit2 + digit1*(n-1-j)
        if mc[0] == '0': continue
        t = int(mc)
        if isprime(t): okset.add(t)
  return len(okset)
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Apr 21 2021

a(n) = A265733(n) + A266141(n) + A266142(n) + A266143(n) + A266144(n) + A266145(n) + A266146(n) + A266147(n) + A266148(n) for n>2.

cp's OEIS Frontend

A266148 Number of n-digit primes in which n-1 of the digits are 9's.

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A266142 Number of n-digit primes in which n-1 of the digits are 3's.

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A266146 Number of n-digit primes in which n-1 of the digits are 7's.

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A266141 Number of n-digit primes in which n-1 of the digits are 2's.

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A266143 Number of n-digit primes in which n-1 of the digits are 4's.

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A266144 Number of n-digit primes in which n-1 of the digits are 5's.

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A266145 Number of n-digit primes in which n-1 of the digits are 6's.

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