A065083 -id:A065083 - cp's OEIS frontend

A105992 Near-repunit primes.

101, 113, 131, 151, 181, 191, 211, 311, 811, 911, 1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111, 10111, 11113, 11117, 11119, 11131, 11161, 11171, 11311, 11411, 16111, 101111, 111119, 111121, 111191, 111211, 111611, 112111, 113111, 131111, 311111, 511111

Offset: 1

Shyam Sunder Gupta, Apr 29 2005

According to the prime glossary "a near-repunit prime is a prime all but one of whose digits are 1." This would also include {2, 3, 5, 7, 13, 17, 19, 31, 41, 61 and 71}, but this sequence only lists terms with more than two digits. - M. F. Hasler, Feb 10 2020

			a(2)=113 is a term because 113 is a prime and all digits are 1 except one.

C. Caldwell and H. Dubner, "The near repunit primes 1(n-k-1)01(1k)," J. Recreational Math., 27 (1995) 35-41.
Heleen, J. P., "More near-repunit primes 1(n-k-1)D(1)1(k), D=2,3, ..., 9," J. Recreational Math., 29:3 (1998) 190-195.

T. D. Noe, Table of n, a(n) for n = 1..1000
Chris Caldwell, The Top 20 Near-repdigit Primes
Chris Caldwell, The Prime Glossary, Near-repunit prime

Cf. A000042, A002275, A004022, A046053, A046413, A065074, A034093, A065083, A102782, A118505.

lst = {}; Do[r = (10^n - 1)/9; Do[AppendTo[lst, DeleteCases[Select[FromDigits[Permutations[Append[IntegerDigits[r], d]]], PrimeQ], r]], {d, 0, 9}], {n, 2, 14}]; Sort[Flatten[lst]] (* Arkadiusz Wesolowski, Sep 20 2011 *)

A065074 Near-repunit primes that contain the digit 0.

Original entry on oeis.org

101, 10111, 101111, 11110111, 11111101, 101111111, 101111111111, 111011111111, 111111011111, 111111110111, 111111111101, 11111111101111111, 11111111111111101, 101111111111111111, 111110111111111111, 111111101111111111, 111111111111011111, 111111111111111011

Offset: 1

Robert G. Wilson v, Nov 19 2001

Alois P. Heinz, Table of n, a(n) for n = 1..956 (this replaces an older b-file by Robert G. Wilson v)
Chris Caldwell, Repunit

Cf. A004022, A034093, A065083.

N:= 20: # to get all terms of up to N digits
A:= select(isprime,[seq(seq((10^n-1)/9 - 10^j,j=n-2..1,-1),n=3..N)]); # Robert Israel, Jun 23 2015

f[n_] := Block[{lst = {}, r = (10^(n - 1) - 1)/9}, AppendTo[ lst, Select[ FromDigits[ Permutations[ Append[ IntegerDigits@ r, 0]]], PrimeQ@# && # > 100 &]]; Union@ Flatten@ lst]; Array[f, 18] // Flatten (* Robert G. Wilson v, Jun 22 2015 *)

Name changed by Arkadiusz Wesolowski, Sep 23 2011

A088281 a(1) = 11; for n > 1, palindromic primes in which a single digit is sandwiched between strings of '1's.

Original entry on oeis.org

11, 101, 131, 151, 181, 191, 11311, 11411, 1114111, 1117111, 111181111, 111191111, 1111118111111, 111111151111111, 111111181111111, 111111111161111111111, 11111111111111611111111111111, 111111111111111111131111111111111111111, 11111111111111111111111111911111111111111111111111111

Offset: 0

Amarnath Murthy, Sep 29 2003

For n > 1, near-repunit palindromic primes (or, palindromic terms of A105992). - Lekraj Beedassy, Jun 05 2009

Harvey P. Dale, Table of n, a(n) for n = 0..32
Index to OEIS entries related to primes involving repdigits.

Cf. A088282, A088283, A088284 (analog with string of '3's, '7's resp. '9's).
Cf. A105992 (near-repunit primes), A065074 (which contain the digit 0), A034093 (number of primes by changing one 1 to 0), A065083 (least k for which that = n).
Cf. A164937 (near-repdigit primes); with 2, ..., 9 as repeated digit: A105982, A105981, A105980, A105979, A105978, A105977, A105976, A105975.

Join[{11},Select[Flatten[Table[FromDigits[Join[PadRight[{},n,1],{d},PadRight[{},n,1]]],{n,26},{d,Cases[Range[0,9],Except[1]]}]],PrimeQ]] (* Harvey P. Dale, Nov 04 2024 *)

print1(11); for(L=1,19,for(d=0,9,d!=1 && ispseudoprime(p=10^(2*L+1)\9+(d-1)*10^L) && print1(","p))) \\ M. F. Hasler, Feb 07 2020

More terms from David Wasserman, Aug 03 2005
Offset changed from 0 to 1 by Lekraj Beedassy, Jun 05 2009
Edited by M. F. Hasler, Feb 07 2020

A034093 Number of near-repunit primes that can be formed from (10^k - 1)/9 by changing just one digit from 1 to 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 5, 0, 0, 0, 0, 2, 5, 0, 4, 0, 0, 0, 3, 0, 1, 0, 0, 1, 2, 0, 4, 1, 0, 1, 2, 0, 2, 1, 0, 0, 7, 0, 4, 0, 0, 0, 2, 0, 2, 1, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 1, 3, 0, 1, 0, 0, 1, 3, 0, 3, 0, 0, 1, 1, 0, 1, 0, 0, 0, 2, 0, 3, 0, 0

Offset: 1

Felice Russo

			a(12) = 5 because from (10^12 - 1)/9 = 111111111111, by changing just one digit from 1 to 0, out of the eleven candidates, 111111111101, 111111110111, 111111011111, 111011111111 and 101111111111 are primes.

C. K. Caldwell and H. Dubner, The near repunits primes, J. Rec. Math., Vol. 27(1), 1995, pp. 35-41.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000
Chris Caldwell, Below are all of the 12-digit Near-Repunit primes.
Chris Caldwell, Repunits.

Cf. A004022, A065074, A065083.

a = {}; Do[ p = IntegerDigits[ (10^n - 1)/9 ]; c = 0; Do[ If[ q = FromDigits[ ReplacePart[p, 0, i]]; PrimeQ[q], c++ ], {i, 2, n} ]; a = Append[a, c], {n, 1, 100} ]; a (* Robert G. Wilson v, Nov 19 2001 *)

a(n)=sum(i=1,n-2,ispseudoprime(10^n\9-10^i)) \\ Charles R Greathouse IV, May 01 2012

from sympy import isprime
def a(n):
    Rn = (10**n-1)//9
    return sum(1 for i in range(n-1) if isprime(Rn-10**i))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Nov 04 2023

More terms from Robert G. Wilson v, Nov 19 2001

Edited by N. J. A. Sloane, Oct 02 2008 at the suggestion of R. J. Mathar

A367081 The least k such that exactly n binary near-repunit primes can be formed from 2^k-1 by changing one digit from 1 to 0.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 38, 24, 18, 36, 48, 20, 248, 588, 144, 252, 5520, 168, 7200, 2400, 2850

Offset: 0

Robert Price, Nov 06 2023

Similar to A065083 but using binary repdigits instead of base 10.
Note that as in A065083, the most significant digit/bit is not replaced with a zero in determining a prime.
a(21) > 7800.
a(25) = 11520 and a(n) > 12000 for n in 21..24 and n > 25 using A272143. - Michael S. Branicky, Nov 09 2023

			a(3)=6 because 2^6 - 1 = 111111_2 and
                      1) 111101_2 = 61,
                      2) 111011_2 = 59,
                      3) 101111_2 = 47,
and no other k < 6 yields exactly three primes.

Cf. A002275, A034093, A065074, A065083, A272143.

a(n) = my(k=1); while(sum(i=1, k-2, ispseudoprime(2^k-1-2^i)) != n, k++); k \\ Thomas Scheuerle, Nov 07 2023

from itertools import count
from sympy import isprime
def A367081(n):
    for k in count(1):
        a, c = (1<= n+1:
                break
        if c == n:
            return k # Chai Wah Wu, Nov 11 2023

cp's OEIS Frontend

A105992 Near-repunit primes.

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A065074 Near-repunit primes that contain the digit 0.

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A088281 a(1) = 11; for n > 1, palindromic primes in which a single digit is sandwiched between strings of '1's.

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A034093 Number of near-repunit primes that can be formed from (10^k - 1)/9 by changing just one digit from 1 to 0.

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A367081 The least k such that exactly n binary near-repunit primes can be formed from 2^k-1 by changing one digit from 1 to 0.

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PARI

Python