A065074 -id:A065074 - 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 *)

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

A164937 Near-repdigit primes.

Original entry on oeis.org

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, 1117, 1151, 1171, 1181, 1511

Offset: 1

G. L. Honaker, Jr., Aug 31 2009

Robert G. Wilson v, Table of n, a(n) for n = 1..22172 (first 5000 terms from Arkadiusz Wesolowski)
Chris Caldwell, The Top 20 Near-repdigit Primes
Chris Caldwell, The Prime Glossary, Near-repdigit prime
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015)

Cf. A105992, A105975, A105976, A105977, A105978, A105979, A105980, A105981, A105982, A065074.

lst = {}; Do[If[PrimeQ[n] && SortBy[Tally[IntegerDigits[n]], Last][[-1, -1]] == IntegerLength[n] - 1, AppendTo[lst, n]], {n, 101, 10^3}]; lst (* Arkadiusz Wesolowski, Sep 18 2011 *)
lst = {}; Do[r = (10^n - 1)/9; Do[AppendTo[lst, DeleteCases[Select[FromDigits[Permutations[Append[IntegerDigits[a*r], d]]], PrimeQ], r | 2 | 3 | 5 | 7]], {a, 9}, {d, 0, 9}], {n, 2, 6}]; Sort[Flatten[lst]] (* Arkadiusz Wesolowski, Sep 22 2011 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    for d in count(3):
        ds = set()
        for end in "1379":
            ds.update(int(c*(d-1) + end) for c in "123456789" if c != end)
            for diff in "0123456789":
                if end == diff: continue
                cands = (end*i + diff + end*(d-1-i) for i in range(d-1))
                ds.update(int(t) for t in cands if t[0] != "0")
        yield from sorted(t for t in ds if isprime(t))
print(list(islice(agen(), 52))) # Michael S. Branicky, May 17 2022

Three more terms from Lekraj Beedassy, Dec 06 2009

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

A331863 Numbers k such that R(k) - 10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

Original entry on oeis.org

8, 12, 17, 20, 24, 42, 1124, 1169, 1538, 7902, 27617, 29684

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subsequence of A065074: near-repunit primes that contain the digit 0.
In base 10, R(k) - 10^floor(k/2-1) has ceiling(k/2) digits 1, one digit 0 and again floor(k/2-1) digits 1: for even as well as odd k, there is a digit 0 just right of the middle of the repunit of length k.
No term can be congruent to 1 (mod 3). - Chai Wah Wu, Feb 07 2020
a(13) > 50000. - Michael S. Branicky, Jul 23 2024

			For k = 8,  R(8)  - 10^(4-1) = 11110111 is prime.
For k = 12, R(12) - 10^(6-1) = 111111011111 is prime.
For k = 17, R(12) - 10^(8-1) = 11111111101111111 is prime.

Cf. A002275 (repunits), A011557 (powers of 10), A065074 (near-repunit primes that contain the digit 0), A138148 (Cyclop numbers with digits 0 & 1).

Cf. A331862 (variant with floor(n/2) instead of floor(n/2-1)), A331860 (variant with + (digit 2) instead of - (digit 0)).

for(n=2,9999,isprime(p=10^n\9-10^(n\2-1))&&print1(n","))

a(7)-a(10) from Giovanni Resta, Jan 31 2020

a(11)-a(12) from Michael S. Branicky, Jul 22 2024

A065083 The least k such that precisely n near-repunit primes can be formed from (10^k-1)/9 by changing one digit from 1 to 0.

Original entry on oeis.org

1, 3, 8, 24, 20, 12, 488, 42, 162, 4848, 642, 1682

Offset: 0

Robert G. Wilson v, Nov 19 2001

Least inverse of A034093. - Charles R Greathouse IV, May 01 2012
a(10) = 642 and a(11) = 1682. - Charles R Greathouse IV, May 03 2012
a(>11) > 5000. - Robert Price, Nov 06 2023

			a(5) = 12 because R_12 = (10^12 -1)/9 = 111111111111 and from this number, by changing just one digit from 1 to 0, out of the eleven candidates, 111111111101, 111111110111, 111111011111, 111011111111 and 101111111111 are primes.

Chris Caldwell, Below are all of the 12-digit Near-Repunit primes:
Chris Caldwell, Repunits

Cf. A034093, A002275, A065074.

a = Table[0, {10} ]; Do[p = IntegerDigits[ (10^n - 1)/9]; c = 0; Do[ If[ q = FromDigits[ ReplacePart[p, 0, i]]; PrimeQ[q], c++ ], {i, 2, n} ]; If[ a[[c + 1]] == 0, a[[c + 1]] = n], {n, 1, 400} ]; a

a(n)=my(k=1);while(sum(i=1,k-2,ispseudoprime(10^k\9-10^i)) != n, k++); k \\ Charles R Greathouse IV, May 01 2012

a(6) from Charles R Greathouse IV, May 01 2012
a(9) from Robert Price, Nov 04 2023
a(10)-a(11) from comments and verified by Robert Price, Nov 04 2023

A331862 Numbers n for which R(n) - 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

Original entry on oeis.org

3, 26, 186, 206, 258, 3486, 12602

Offset: 1

M. F. Hasler, Jan 30 2020

The corresponding primes are a subsequence of A065074: near-repunit primes that contain the digit 0.
In base 10, R(n) - 10^floor(n/2) has ceiling(n/2)-1 digits 1, one digit 0, and again floor(n/2) digits 1. For odd n, this is a palindrome, for even n the digit 0 is just left to the middle of the number.
There can't be an odd term > 3 because the corresponding palindrome factors as R((n-1)/2)*(10^((n+1)/2) + 1).
No term can be congruent to 1 mod 3. - Chai Wah Wu, Feb 07 2020

			For n = 3, R(n) - 10^floor(n/2) = 101 is prime.
For n = 26, R(n) - 10^floor(n/2) = 11111111111101111111111111 is prime.

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015)

Cf. A002275 (repunits), A004023 (indices of prime repunits), A011557 (powers of 10), A065074 (near-repunit primes that contain the digit 0), A105992 (near-repunit primes), A138148 (Cyclops numbers with digits 0 & 1).

Cf. A331860 (variant with digit 2 instead of digit 0), A331863 (variant with floor(n/2-1) instead of floor(n/2)).

for(n=0,9999,isprime(p=10^n\9-10^(n\2))&&print1(n","))

a(6)-a(7) from Giovanni Resta, Jan 31 2020

A263431 Near-repdigit primes with only digits 9 and a single 8 in decimal expansion.

Original entry on oeis.org

89, 8999, 98999, 99989, 989999, 9899999, 89999999, 99899999, 99998999, 99999989, 998999999, 98999999999, 99989999999, 999998999999, 999999999899, 999999999989, 99899999999999, 99999899999999, 99999999899999, 999999899999999, 999999999989999, 999999999999989

Offset: 1

Felix Fröhlich, Oct 18 2015

Felix Fröhlich, Table of n, a(n) for n = 1..200

Cf. A020472, A065074, A069661, A105975, A178007.

Select[Flatten[Table[FromDigits/@Permutations[PadRight[{8},n,9]],{n,15}]],PrimeQ] (* Harvey P. Dale, Mar 29 2020 *)

a002283(n) = 10^n-1
a011557(n) = 10^n
num(n, k) = a002283(n)-a011557(k)
terms(n) = i=0; x=1; while(x > 0, y=x-1; while(y >= 0, if(ispseudoprime(num(x, y)), print1(num(x, y), ", "); i++); if(i==n, break({2})); y--); x++)
terms(30) \\ print initial thirty terms

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

A168340 Near-repunit emirps with a single 0 in their decimal expansion.

Original entry on oeis.org

101111111111, 111011111111, 111111110111, 111111111101, 111110111111111111, 111111111111011111, 11011111111111111111, 11111111111111111011, 111111111111111111111101111111111111111111111111111111, 111111111111111111111111111111101111111111111111111111

Offset: 1

Lekraj Beedassy, Nov 23 2009

Emirps A006567 which are also "near" repunits (near indicating that one non-leading 1-digit is replaced by a 0).

Cf. A065074.

A065074 INTERSECT A006567. [R. J. Mathar, Nov 26 2009]

6 more terms from R. J. Mathar, Nov 26 2009

Name changed by Arkadiusz Wesolowski, Sep 23 2011

cp's OEIS Frontend

A105992 Near-repunit primes.

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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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A164937 Near-repdigit primes.

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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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A331863 Numbers k such that R(k) - 10^floor(k/2-1) is prime, where R(k) = (10^k-1)/9 (repunit: A002275).

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A065083 The least k such that precisely n near-repunit primes can be formed from (10^k-1)/9 by changing one digit from 1 to 0.

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A331862 Numbers n for which R(n) - 10^floor(n/2) is prime, where R(n) = (10^n-1)/9.

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A263431 Near-repdigit primes with only digits 9 and a single 8 in decimal expansion.