A002275 -id:A002275 - cp's OEIS frontend

A056654 Numbers k such that 10*R_k + 3 is prime, where R_k is the repunit (A002275) of length k.

0, 1, 2, 4, 8, 10, 23, 83, 220, 1313, 2951, 20015, 51053

Offset: 1

Robert G. Wilson v, Aug 09 2000

Also numbers k such that (10^(k+1)+17)/9 is prime.

a(14) > 10^5. - Robert Price, Nov 01 2014

			8 is a term because 111111113 is a prime.

Makoto Kamada, Prime numbers of the form 11...113.
Index entries for primes involving repunits

Cf. A093011 (corresponding primes), A097683.

Do[ If[ PrimeQ[ 10*(10^n - 1)/9 + 3 ], Print[ n ] ], {n, 0, 1350} ]

is(n)=ispseudoprime(10^n\9*10+3) \\ Charles R Greathouse IV, Nov 10 2021

a(n) = A097683(n+1) - 1. - Robert Price, Nov 01 2014

a(11) (only a probable prime) from Rick L. Shepherd, Mar 14 2004

a(12)-a(13) derived from A097683 by Robert Price, Nov 01 2014

A242614 Triangle read by rows: row n contains numbers with sum of digits = n, and not greater than the n-th repunit (cf. A007953 and A002275).

Original entry on oeis.org

0, 1, 2, 11, 3, 12, 21, 30, 102, 111, 4, 13, 22, 31, 40, 103, 112, 121, 130, 202, 211, 220, 301, 310, 400, 1003, 1012, 1021, 1030, 1102, 1111, 5, 14, 23, 32, 41, 50, 104, 113, 122, 131, 140, 203, 212, 221, 230, 302, 311, 320, 401, 410, 500, 1004, 1013, 1022

Offset: 0

Reinhard Zumkeller, Jul 16 2014

Number of terms in row n = A242622(n);
T(n,1) = A051885(n);
T(n,A242622(n)) = A002275(n);
for n > 0: number of repdigit terms in row n = A242627(n).

			The triangle begins:
. 0:  0
. 1:  1
. 2:  2,11
. 3:  3,12,21,30,102,111
. 4:  4,13,22,31,40,103,112,121,130,202, . . . ,1021,1030,1102,1111
. 5:  5,14,23,32,41,50,104,113,122,131, . . . ,11021,11030,11102,11111 .

Reinhard Zumkeller, Rows n = 0..8 of table, flattened

Cf. A007953 (sum of digits), A002275 (repunits).

Cf. A011557, A052216, A052217, A052218, A052219, A052220, A052221, A052222, A052223, A052224, A166311, A235151, A143164, A235225, A235226, A235227, A166370, A235228, A166459, A235229.

a242614 n k = a242614_row n !! (k-1)
a242614_row n = filter ((== n) . a007953) [n .. a002275 n]
a242614_tabf = map a242614_row [0..]

Join[{0},Flatten[Table[Select[Range[FromDigits[PadRight[{},n,1]]], Total[ IntegerDigits[ #]] == n&],{n,5}]]] (* Harvey P. Dale, Oct 08 2019 *)

A003020 Largest prime factor of the "repunit" number 11...1 (cf. A002275).

Original entry on oeis.org

11, 37, 101, 271, 37, 4649, 137, 333667, 9091, 513239, 9901, 265371653, 909091, 2906161, 5882353, 5363222357, 333667, 1111111111111111111, 27961, 10838689, 513239, 11111111111111111111111, 99990001, 182521213001, 1058313049

Offset: 2

N. J. A. Sloane

a(n) = R_n iff n is a term of A004023. - Bernard Schott, Jul 07 2022

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Introduction à la Théorie des Nombres. Gauthier-Villars, Paris, 1952, p. 40.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Factors of the Repunits 11 through R_40, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1986, p. 219.

Max Alekseyev, Table of n, a(n) for n = 2..352 (terms a(2)..a(100) from T. D. Noe, derived from data from Yousuke Koide; a(101)..a(322) from Ray Chandler)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Patrick De Geest, Repunits and their Prime Factors
Makoto Kamada, Factorizations of 11...11 (Repunit).
Yousuke Koide, Factorizations of Repunit Numbers
A. A. D. Steward, Factorization of Repunits[up to R(196)] [Broken link]
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A002275, A004023, A006530, A102380.
Same as A005422 except for initial terms.
Smallest factor: A067063.

Table[Max[Transpose[FactorInteger[10^i - 1]][[1]]], {i, 2, 25}]
Table[FactorInteger[FromDigits[PadRight[{},n,1]]][[-1,1]],{n,2,30}] (* Harvey P. Dale, Feb 01 2014 *)

a(n)=local(p); if(n<2,n==1,p=factor((10^n-1)/9)~[1,]; p[length(p)])

a(n) = A006530(A002275(n)). - Ray Chandler, Apr 22 2017

More terms from Harvey P. Dale, Jan 17 2001

A096508 Numbers k for which 8*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

2, 14, 17, 35, 4175, 4472, 9812, 12260, 12341, 13760, 14576, 53411, 144683, 148328

Offset: 1

Labos Elemer, Jul 12 2004

nonn

Also numbers k such that (8*10^k + 1)/9 is prime.

a(15) > 2*10^5. - Robert Price, Sep 06 2014

			35 is a term because 88888888888888888888888888888888889 (34 8's) is a prime number.

Makoto Kamada, Prime numbers of the form 88...889.
Index entries for primes involving repunits

Cf. A002275, A056663, A096503-A096507.

select(n -> isprime((8*10^n+1)/9), [$1..10000]); # Robert Israel, Sep 07 2014

Do[ If[ PrimeQ[ 8(10^n - 1)/9 + 1], Print[n]], {n, 0, 30000}] (* Robert G. Wilson v, Oct 15 2004 *)

for(n=1,10^4,if(ispseudoprime(8*(10^n-1)/9+1),print1(n,", "))) \\ Derek Orr, Sep 06 2014

a(n) = A056663(n) + 1.

Four missing terms (9812, 12260, 12341, 13760) added, and a(12)-a(14) added from Kamada data, by Robert Price, Sep 06 2014

A095714 Numbers k such that 9*R_k - 8 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

3, 5, 7, 33, 45, 105, 197, 199, 281, 301, 317, 1107, 1657, 3395, 35925, 37597, 64305, 80139, 221631

Offset: 1

Alonso del Arte, Jul 07 2004

nonn

Also numbers k such that 10^k - 9 is a prime.

			a(2) = 5, since 10^5 - 9 = 99991, which is prime.

Makoto Kamada, Prime numbers of the form 99...991.
Index entries for primes involving repunits

Cf. A002275, A093177, A056696.

Do[ If[ PrimeQ[10^n - 9], Print[n]], {n, 0, 7000}]

a(n) = A056696(n) + 1.

a(12) - a(14) from Robert G. Wilson v, Oct 15 2004
a(15) - a(16) from Jason Earls, Jan 07 2008
a(17) - a(19) from Alexander Gramolin, May 13 2011
Edited by Ray Chandler, Feb 26 2012
Title corrected by Robert Price, Sep 06 2014

A102380 Irregular triangle read by rows in which row n lists prime factors (with multiplicity) of the repunit (10^n - 1)/9 (A002275(n)).

Original entry on oeis.org

1, 11, 3, 37, 11, 101, 41, 271, 3, 7, 11, 13, 37, 239, 4649, 11, 73, 101, 137, 3, 3, 37, 333667, 11, 41, 271, 9091, 21649, 513239, 3, 7, 11, 13, 37, 101, 9901, 53, 79, 265371653, 11, 239, 4649, 909091, 3, 31, 37, 41, 271, 2906161, 11, 17, 73, 101

Offset: 1

N. J. A. Sloane, Nov 28 2006

See A003020 for other links and references.

			First rows:
    1;
   11;
    3,   37;
   11,  101;
   41,  271;
    3,    7, 11, 13, 37;
  239, 4649;
  ...

Rows n=1..352, flattened
Makoto Kamada, Factorizations of 11...11 (repunit).
Yousuke Koide, Factorizations of Repunit Numbers
Franz Lemmermeyer, Simerka - Quadratic Forms and Factorization, arXiv:1201.0282 [math.NT], 2011. For the 19th row = [2071723, 5363222357].
Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Cf. A002275, A067063, A003020, A204845, A204846, A204847, A204848.

Maple
```
[seq( ifactor((10^n-1)/9),n=1..20)];
```

First 100 rows in b-file from T. D. Noe, Feb 27 2009
Rows n=101..322 in b-file from Ray Chandler, May 01 2017
Rows n=323..352 in b-file from Max Alekseyev, Apr 26 2022

A067063 Smallest prime factor of repunit(n) = (10^n-1)/9 (A002275).

Original entry on oeis.org

11, 3, 11, 41, 3, 239, 11, 3, 11, 21649, 3, 53, 11, 3, 11, 2071723, 3, 1111111111111111111, 11, 3, 11, 11111111111111111111111, 3, 41, 11, 3, 11, 3191, 3, 2791, 11, 3, 11, 41, 3, 2028119, 11, 3, 11, 83, 3, 173, 11, 3, 11, 35121409, 3, 239

Offset: 2

Amarnath Murthy, Jan 03 2002

nonn

a(n) = A003020(n) = R_(n) iff n is a term of A004023. - Bernard Schott, May 22 2022

David Wells, The Penguin Dictionary of Curious and Interesting Numbers.

Ray Chandler, Table of n, a(n) for n = 2..508 (first 499 terms from T. D. Noe - corrected 7 terms)
Makoto Kamada, Factorizations of 11...11 (Repunit).
Yousuke Koide, Factorizations of Repunit Numbers
Amarnath Murthy, On the divisors of Smarandache Unary Sequence, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000, page 184.
Samuel S. Wagstaff, the Cunningham Project

Cf. A002275, A004023, A020639, A102380.

Largest factor: A003020.

'min(op(numtheory[factorset]((10^k-1)/9)))'$k=2..50; # M. F. Hasler, Nov 21 2006

a = {}; Do[a = Append[a, FactorInteger[(10^n - 1)/9][[1, 1]]], {n, 2, 111} ]; a
Table[FactorInteger[FromDigits[PadRight[{},n,1]]][[1,1]],{n,2,50}] (* Harvey P. Dale, Dec 10 2013 *)

a(3n) = 3, a(6n-4) = a(6n-2) = 11, a(30n-25) = a(30n-5) = 41, ... - M. F. Hasler, Nov 21 2006

a(n) = A020639(A002275(n)). - Ray Chandler, Apr 22 2017

More terms from Robert G. Wilson v, Jan 04 2002

A096507 Numbers k such that 6*R_k + 1 is a prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

1, 2, 6, 8, 9, 11, 20, 23, 41, 63, 66, 119, 122, 149, 252, 284, 305, 592, 746, 875, 1204, 1364, 2240, 2403, 5106, 5776, 5813, 12456, 14235, 39606, 55544, 84239, 275922

Offset: 1

Labos Elemer, Jul 12 2004

nonn

Also numbers k such that (2*10^k + 1)/3 is prime.
These numbers form a near-repdigit sequence (6)w7.
All the terms from k = 2403 through 14235 correspond to primes. - Joao da Silva (zxawyh66(AT)yahoo.com), Oct 03 2005

			k = 9 gives 2000000001/3 = 666666667, which is prime.
k = 20 gives 66666666666666666667, which is prime.

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

Cf. A002275, A056657, A093170, A096503, A096504, A096505, A096506, A096508.

Select[Range@ 2500, PrimeQ[FromDigits@ Table[6, {#}] + 1] &] (* or *)
Select[Range@ 2500, PrimeQ[2 (10^# - 1)/3 + 1] &] (* Michael De Vlieger, Jul 04 2016 *)

a(n) = A056657(n) + 1.

More terms from Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004
39606 and 55544 from Serge Batalov, Jun 2009
84239 from Serge Batalov, Jul 06 2009 confirmed as next term by Ray Chandler, Feb 23 2012
a(33) from Kamada data by Tyler Busby, Apr 14 2024

A100706 Bisection of A002275.

Original entry on oeis.org

1, 111, 11111, 1111111, 111111111, 11111111111, 1111111111111, 111111111111111, 11111111111111111, 1111111111111111111, 111111111111111111111, 11111111111111111111111

Offset: 0

N. J. A. Sloane, Nov 19 2004

Also the binary representation of the n-th iteration of the elementary cellular automaton starting with a single ON (black) cell for Rules 151, 159, 183, 191, 215, 222, 223, 247, 254 and 255. - Robert Price, Feb 21 2016

The aerated sequence 1, 0, 111, 0, 11111, 0, 1111111, ... is a linear divisibility sequence of order 4. It is the case P1 = 0, P2 = -9^2, Q = -10 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. Cf. A007583, A095372 and A299960. - Peter Bala, Aug 28 2019

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (101,-100).

Cf. A002275, A099814 (other bisection), A007583, A095372, A299960.

seq((10^(2*n+1) - 1)/9,n=0..15); # C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

Table[(10^(2*n + 1) - 1)/9, {n, 0, 100}] (* Robert Price, Feb 21 2016 *)

a(n) = (10^(2*n + 1) - 1)/9; \\ Michel Marcus, Mar 12 2023

def A100706(n): return (10**((n<<1)+1)-1)//9 # Chai Wah Wu, Nov 04 2022

Numbers composed entirely of 2*n+1 concatenated 1's for n >= 0.
O.g.f.: (1+10*x)/((-1+x)*(-1+100*x)). - R. J. Mathar, Apr 03 2008
From Klaus Purath, Sep 23 2020: (Start)
a(n) = Sum_{i = 0..2*n} 10^i.
a(n) = 101*a(n-1) - 100*a(n-2).
a(n) = 110*10^(2*n-2) + a(n-1).
a(n) = 100*a(n-1) + 11.
a(n) = (a(n-1)^2 - 1210*10^(2*n-4))/a(n-2). (End)

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

A272525 Convolution of nonzero repunits (A002275) with themselves.

Original entry on oeis.org

1, 22, 343, 4664, 58985, 713306, 8367627, 96021948, 1083676269, 12071330590, 133058984911, 1454046639232, 15775034293553, 170096021947874, 1824417009602195, 19478737997256516, 207133058984910837, 2194787379972565158, 23182441700960219479, 244170096021947873800

Offset: 0

Ilya Gutkovskiy, May 02 2016

Partial sums of A014925.

Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (22,-141,220,-100)

Cf. A002275, A010879, A014925, A083449.

LinearRecurrence[{22, -141, 220, -100}, {1, 22, 343, 4664}, 20]
Table[(9 n (10^(n + 2) + 1) + 7 10^(n + 2) + 29)/729, {n, 0, 19}]

A272525(n)=(9*n+7)*(10^(n+2)+1)\729+1 \\ M. F. Hasler, Nov 02 2016

O.g.f.: 1/((1 - 10*x)^2*(1 - x)^2).
E.g.f.: (29 + 9*x  + 700*exp(9*x) + 9000*x*exp(9*x))*exp(x)/729.
a(n) =  22*a(n-1) - 141*a(n-2) + 220*a(n-3) - 100*a(n-4).
a(n) = (9*n(10^(n+2) + 1) + 7*10^(n+2) + 29)/729.
A010879(a(n)) = A010879(n+1).

cp's OEIS Frontend

A056654 Numbers k such that 10*R_k + 3 is prime, where R_k is the repunit (A002275) of length k.

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A242614 Triangle read by rows: row n contains numbers with sum of digits = n, and not greater than the n-th repunit (cf. A007953 and A002275).

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A003020 Largest prime factor of the "repunit" number 11...1 (cf. A002275).

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A096508 Numbers k for which 8*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A095714 Numbers k such that 9*R_k - 8 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

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A102380 Irregular triangle read by rows in which row n lists prime factors (with multiplicity) of the repunit (10^n - 1)/9 (A002275(n)).

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A067063 Smallest prime factor of repunit(n) = (10^n-1)/9 (A002275).

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