A002275 -id:A002275 - cp's OEIS frontend

A097683 Numbers k such that R_k + 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

0, 1, 2, 3, 5, 9, 11, 24, 84, 221, 1314, 2952, 20016, 51054

Offset: 1

Carl R. White and Julien Peter Benney (jpbenney(AT)ftml.net), Aug 19 2004

Also numbers k such that (10^k + 17)/9 is prime.
The corresponding values R_k + 2 are primes of the form "(n-1) ones followed by a three"; zero is a degenerate case. Related to the base-10 repunit primes.
a(15) > 10^5. - Robert Price, Oct 12 2014
By Kamada link, a(15) > 4*10^5. - Jeppe Stig Nielsen, Jan 17 2023

			11113 = ((10^5)+17)/9 and 11113 is prime.

Makoto Kamada, Prime numbers of the form 11...113.
Henri Lifchitz and Renaud Lifchitz, PRP Top, Search output for (10^k+17)/9
Index entries for primes involving repunits

Cf. A002275, A004023, A056654, A097684, A097685.

A097683:=n->`if`((10^n+17 mod 9) = 0 and isprime(floor((10^n+17)/9)),n,NULL): seq(A097683(n), n=0..10^3); # Wesley Ivan Hurt, Oct 12 2014

Do[ If[ PrimeQ[(10^n - 1)/9 + 2], Print[n]], {n, 0, 5951}] (* Robert G. Wilson v, Oct 15 2004 *)

a(n) = A056654(n-1) + 1.

a(11)-a(12) from Robert G. Wilson v, Oct 15 2004
Edited by N. J. A. Sloane, Apr 02 2009, at the suggestion of Farideh Firoozbakht
a(13) from Kamada link by Ray Chandler, Dec 23 2010
a(14) from Robert Price, Oct 12 2014

A256077 Repeat 2^d times the repunit A002275(d); d = 1, 2, 3...

Original entry on oeis.org

1, 1, 11, 11, 11, 11, 111, 111, 111, 111, 111, 111, 111, 111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 1111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111, 11111

Offset: 1

M. F. Hasler, Mar 21 2015

nonn

Yields the length of the n-th (nonempty) binary word (or word over any 2-letter alphabet, like A007931 or A032810 or A032834) in tally mark notation (A000042).

Felix Fröhlich, Table of n, a(n) for n = 1..10000

lim = 5; lst = Table[(10^n - 1)/9, {n, 0, lim}]; Reap@ For[i = 1, i <= lim, i++, Sow@ Table[lst[[i + 1]], {d, 2^i}]] // Flatten // Rest (* Michael De Vlieger, Apr 01 2015 *)

PARI
```
a(n)=10^#binary(n+1)\90
```

def A256077(n): return (10**((n+1).bit_length()-1)-1)//9 # Chai Wah Wu, Nov 07 2024

a(n) = A002275(A000523(n+1)) = A032810(n)-A007931(n) = A032834(n)-A032810(n), etc.

A056657 Numbers k such that 60*R_k + 7 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

0, 1, 5, 7, 8, 10, 19, 22, 40, 62, 65, 118, 121, 148, 251, 283, 304, 591, 745, 874, 1203, 1363, 2239, 2402, 5105, 5775, 5812, 12455, 14234, 39605, 55543, 84238, 275921

Offset: 1

Robert G. Wilson v, Aug 09 2000

nonn

Also numbers k such that (20*10^k+1)/3 is prime.

			7, 67, 666667, 66666667, 666666667, 66666666667, etc. are primes.

Makoto Kamada, Prime numbers of the form 66...667.
Index entries for primes involving repunits

Cf. A002275, A093170, A096507.

Do[ If[ PrimeQ[ 60*(10^n - 1)/9 + 7 ], Print[n]], {n, 25556}]

a(n) = A096507(n) - 1.

More terms from Robert G. Wilson v, Oct 22 2003
2239,2402,5105,5775 from Farideh Firoozbakht, Dec 23 2003
39605 and 55543 from Serge Batalov, Jun 2009
84238 from Serge Batalov, Jul 06 2009 confirmed as next term by Ray Chandler, Feb 23 2012
a(33) derived from A096507 by Robert Price, Jul 07 2024

A093671 Primes of the form 10^k + 3*R_k, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

13, 1333333333333333, 133333333333333333333333333333333333333333, 133333333333333333333333333333333333333333333333333333333333333333333333333333333333

Offset: 1

Rick L. Shepherd, Apr 08 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..9
Makoto Kamada, Prime numbers of the form 133...33.
Index entries for primes involving repunits

Cf. A002275 (repunits), A056698 (corresponding k).

[a: n in [0..100] | IsPrime(a) where a is ((4*10^n-1) div 3)]; // Vincenzo Librandi, Apr 06 2019

Select[Table[FromDigits[PadRight[{1}, n, 3]], {n, 500}], PrimeQ] (* Vincenzo Librandi, Apr 06 2019 *)

A096506 Numbers n for which 2*R_n + 1 is a prime, where R_n = 11...1 is the repunit (A002275) of length n.

Original entry on oeis.org

1, 2, 3, 8, 11, 36, 95, 101, 128, 260, 351, 467, 645, 1011, 1178, 1217, 2442, 3761, 3806, 15617, 26459, 63117, 88545, 93497

Offset: 1

Labos Elemer, Jul 12 2004

nonn

Also numbers n such that (2*10^n + 7)/9 is prime.

Per Kamada link, 181457, 202059, 262874 are also terms, found by Rytis Slatkevicius. - Michael S. Branicky, Sep 13 2024

			n=36: 222222222222222222222222222222222223 is a prime number.

Makoto Kamada, Prime numbers of the form 22...223.
Index entries for primes involving repunits

Cf. A056656, A093162, A096503, A096504, A096505, A096507, A096508.

Do[ If[ PrimeQ[ 2(10^n - 1)/9 + 1], Print[n]], {n, 7000}] (* Robert G. Wilson v, Oct 14 2004 *)

a(n) = A056656(n) + 1.

a(20)-a(24) from Kamada link by Ray Chandler, Feb 27 2012

A204845 Irregular triangle read by rows in which row n lists primitive prime factors of the repunit (10^n - 1)/9 (A002275(n)).

Original entry on oeis.org

1, 11, 3, 37, 101, 41, 271, 7, 13, 239, 4649, 73, 137, 333667, 9091, 21649, 513239, 9901, 53, 79, 265371653, 909091, 31, 2906161, 17, 5882353, 2071723, 5363222357, 19, 52579, 1111111111111111111, 3541, 27961, 43, 1933, 10838689, 23, 4093, 8779, 11111111111111111111111

Offset: 1

N. J. A. Sloane, Jan 19 2012

			Triangle begins:
1
11
3 37
101
41 271
7 13
239 4649
73 137
333667
9091
...

Ray Chandler, Rows n = 1..322, flattened (first 60 rows from Alois P. Heinz)
Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Cf. A002275, A102380, A204846, A204847, A204848.

with(numtheory):
S:= proc(n) option remember;
      `if`(n=1, {1}, S(n-1) union factorset ((10^n-1)/9))
    end:
T:= n-> sort([(S(n) minus `if`(n=1, {}, S(n-1)))[]])[]:
seq(T(n), n=1..30);  # Alois P. Heinz, Feb 17 2012

S[n_] := S[n] = If[n==1, {1}, S[n-1] ~Union~ FactorInteger[(10^n-1)/9][[ All, 1]]]; T[n_] := Sort[S[n] ~Complement~ If[n==1, {}, S[n-1]]]; Table[ T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Mar 13 2017, after Alois P. Heinz *)

More terms from Alois P. Heinz, Feb 17 2012

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

A084832 Numbers k such that 2*R_k - 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

4, 18, 100, 121, 244, 546, 631, 1494, 2566, 8088, 262603, 282948, 359860

Offset: 1

Jason Earls, Jun 05 2003

Also numbers k such that (2*10^k-11)/9 is prime.
Larger values correspond to strong pseudoprimes.
a(11) > 10^5. - Robert Price, Sep 06 2014

			a(1) = 4 because 2*(10^4-1)/9-1 = 2221 is prime.
a(2) = 18 means that 222222222222222221 is prime.

Makoto Kamada, Prime numbers of the form 22...221.
Index entries for primes involving repunits

Cf. A084831, A096503-A096508, A096841-A096846, A002275, A056660.

select(t -> isprime(2*(10^t-1)/9-1),[$1..1000]); # Robert Israel, Sep 07 2014

Do[ If[ PrimeQ[2(10^n - 1)/9 - 1], Print[n]], {n, 0, 7000}] (* Robert G. Wilson v, Oct 14 2004; fixed by Derek Orr, Sep 06 2014 *)

for(n=1, 10^4, if(ispseudoprime(2*(10^n-1)/9-1), print1(n,", "))) \\ Derek Orr, Sep 06 2014

from sympy import isprime
def afind(limit):
  n, twoRn = 1, 2
  for n in range(1, limit+1):
    if isprime(twoRn-1): print(n, end=", ")
    twoRn = 10*twoRn + 2
afind(700) # Michael S. Branicky, Apr 18 2021

a(n) = A056660(n) + 1.

a(8) from Labos Elemer, Jul 15 2004
a(10) from Kamada data by Robert Price, Sep 06 2014
a(11)-a(13) from Kamada data by Tyler Busby, Apr 29 2024

A093174 Primes of the form 40*R_k + 1, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

41, 4441, 44444444441, 4444444444444444444444444441, 4444444444444444444444444444444444444444444444444444441, 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444441

Offset: 1

Rick L. Shepherd, Mar 27 2004

nonn

Primes of the form (4*10^k - 31)/9. - Vincenzo Librandi, Nov 16 2010

Makoto Kamada, Prime numbers of the form 44...441.
Index entries for primes involving repunits

Cf. A056681 (corresponding k), A173768.

Edited by N. J. A. Sloane, Feb 26 2010

A204846 Irregular triangle read by rows in which row n lists algebraic prime factors of the repunit (10^n - 1)/9 (A002275(n)).

Original entry on oeis.org

1, 1, 1, 11, 1, 3, 11, 37, 1, 11, 101, 3, 37, 11, 41, 271, 1, 3, 7, 11, 13, 37, 101, 1, 11, 239, 4649, 3, 37, 41, 271, 11, 73, 101, 137, 1, 3, 7, 11, 13, 37, 333667, 1, 11, 41, 101, 271, 9091, 3, 37, 239, 4649, 11, 21649, 513239, 1, 3, 7, 11, 13, 37, 73, 101, 137, 9901, 41, 271

Offset: 1

N. J. A. Sloane, Jan 19 2012

			Triangle begins:
  1;
  1;
  1;
  11;
  1;
  3,11,37;
  1;
  11,101;
  3,37;
  11,41,271;
  ...

Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Cf. A002275, A102380, A204845.

rows[nmax_] := Module[{prim = {1}, r = {{1}}, p, c}, Do[p = FactorInteger[(10^n - 1)/9][[;; , 1]]; c = Complement[p, Complement[p, prim]]; If[c == {}, c = {1}]; AppendTo[r, c]; prim = Union[prim, p], {n, 2, nmax}]; r]; rows[25] // Flatten (* Amiram Eldar, May 11 2024 *)

More terms from Amiram Eldar, May 11 2024

cp's OEIS Frontend

A097683 Numbers k such that R_k + 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A256077 Repeat 2^d times the repunit A002275(d); d = 1, 2, 3...

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A056657 Numbers k such that 60*R_k + 7 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A093671 Primes of the form 10^k + 3*R_k, where R_k is the repunit (A002275) of length k.

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Magma

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A096506 Numbers n for which 2*R_n + 1 is a prime, where R_n = 11...1 is the repunit (A002275) of length n.

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A204845 Irregular triangle read by rows in which row n lists primitive prime factors of the repunit (10^n - 1)/9 (A002275(n)).

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Maple

Mathematica

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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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A084832 Numbers k such that 2*R_k - 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.