A102380 -id:A102380 - cp's OEIS frontend

A002275 Repunits: (10^n - 1)/9. Often denoted by R_n.

0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111, 11111111111111111111

Offset: 0

N. J. A. Sloane

R_n is a string of n 1's.
Base-4 representation of Jacobsthal bisection sequence A002450. E.g., a(4)= 1111 because A002450(4)= 85 (in base 10) = 64 + 16 + 4 + 1 = 1*(4^3) + 1*(4^2) + 1*(4^1) + 1. - Paul Barry, Mar 12 2004
Except for the first two terms, these numbers cannot be perfect squares, because x^2 != 11 (mod 100). - Zak Seidov, Dec 05 2008
For n >= 0: a(n) = (A000225(n) written in base 2). - Jaroslav Krizek, Jul 27 2009, edited by M. F. Hasler, Jul 03 2020
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
Except 0, 1 and 11, all these integers are Brazilian numbers, A125134. - Bernard Schott, Dec 24 2012
Numbers n such that 11...111 = R_n = (10^n - 1)/9 is prime are in A004023. - Bernard Schott, Dec 24 2012
The terms 0 and 1 are the only squares in this sequence, as a(n) == 3 (mod 4) for n>=2. - Nehul Yadav, Sep 26 2013
For n>=2 the multiplicative order of 10 modulo the a(n) is n. - Robert G. Wilson v, Aug 20 2014
The above is a special case of the statement that the order of z modulo (z^n-1)/(z-1) is n, here for z=10. - Joerg Arndt, Aug 21 2014
From Peter Bala, Sep 20 2015: (Start)
Let d be a divisor of a(n). Let m*d be any multiple of d. Split the decimal expansion of m*d into 2 blocks of contiguous digits a and b, so we have m*d = 10^k*a + b for some k, where 0 <= k < number of decimal digits of m*d. Then d divides a^n - (-b)^n (see McGough). For example, 271 divides a(5) and we find 2^5 + 71^5 = 11*73*271*8291 and 27^5 + 1^5 = 2^2*7*31*61*271 are both divisible by 271. Similarly, 4*271 = 1084 and 10^5 + 84^5 = 2^5*31*47*271*331 while 108^5 + 4^5 = 2^12*7*31*61*271 are again both divisible by 271. (End)
Starting with the second term this sequence is the binary representation of the n-th iteration of the Rule 220 and 252 elementary cellular automaton starting with a single ON (black) cell. - Robert Price, Feb 21 2016
If p > 5 is a prime, then p divides a(p-1). - Thomas Ordowski, Apr 10 2016
0, 1 and 11 are only terms that are of the form x^2 + y^2 + z^2 where x, y, z are integers. In other words, a(n) is a member of A004215 for all n > 2. - Altug Alkan, May 08 2016
Except for the initial terms, the binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 17 2017
The term "repunit" was coined by Albert H. Beiler in 1964. - Amiram Eldar, Nov 13 2020
q-integers for q = 10. - John Keith, Apr 12 2021
Binomial transform of A001019 with leading zero. - Jules Beauchamp, Jan 04 2022

Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, New York: Dover Publications, 1964, chapter XI, p. 83.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 235-237.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 197-198.
Samuel Yates, Peculiar Properties of Repunits, J. Recr. Math. 2, 139-146, 1969.
Samuel Yates, Prime Divisors of Repunits, J. Recr. Math. 8, 33-38, 1975.

David Wasserman, Table of n, a(n) for n = 0..1000
Eudes Antonio Costa and Fernando Soares Carvalho, On repunit polynomials sequence, Braz. Elec. J. Math. (2024). See pp. 2, 15.
Eudes Antonio Costa, Douglas Catulio Santos, Paula Maria Machado Cruz Catarino, and Elen Viviani Pereira Spreafico, On Gaussian and Quaternion Repunit Numbers, Rev. Mat. UFOP (Brazil, 2024) Vol. 2. See p. 2.
Eudes Antonio Costa, Paula Maria Machado Cruz Catarino, and Douglas Catulio Santos, A Study of the Symmetry of the Tricomplex Repunit Sequence with Repunit Sequence, Symmetry (2024) Vol. 17, No. 1, 28.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 3, 18.
Makoto Kamada, Factorizations of 11...11 (Repunit).
Douglas Catulio Santos, Eudes Antonio Costa, and Paula Maria Machado Cruz Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 4.
W. M. Snyder, Factoring Repunits, Am. Math. Monthly, Vol. 89, No. 7 (1982), pp. 462-466.
Amelia Carolina Sparavigna, On Repunits, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Repunit.
Eric Weisstein's World of Mathematics, Demlo Number.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Wikipedia, Repunit.
Amin Witno, A Family of Sequences Generating Smith Numbers, J. Int. Seq. 16 (2013) #13.4.6.
Stephen Wolfram, A New Kind of Science.
Samuel Yates, The Mystique of Repunits, Math. Mag., Vol. 51, No. 1 (1978), pp. 22-28.
Index to Elementary Cellular Automata.
Index entries for 10-automatic sequences.
Index entries for sequences related to cellular automata.
Index entries for linear recurrences with constant coefficients, signature (11,-10).
Index entries for "core" sequences.

Partial sums of 10^n (A011557). Factors: A003020, A067063.
Bisections give A099814, A100706.
Cf. A000042, A002276, A002277, A002278, A002279, A002280, A002281, A002282, A002283, A004023, A046053, A059988, A065444, A075412, A075415, A083278, A095370, A102380, A125134, A178635, A204845, A204846, A204847, A204848, A206244.
Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.
Cf. A000225, A001019, A001045, A002450, A004215, A015565, A125118, A242614, A242622.
Cf. A010785, A017173, A105279.

a002275 = (`div` 9) . subtract 1 . (10 ^)
a002275_list = iterate ((+ 1) . (* 10)) 0
-- Reinhard Zumkeller, Jul 05 2013, Feb 05 2012

[(10^n-1)/9: n in [0..25]]; // Vincenzo Librandi, Nov 06 2014

seq((10^k - 1)/9, k=0..30); # Wesley Ivan Hurt, Sep 28 2013

Table[(10^n - 1)/9, {n, 0, 19}] (* Alonso del Arte, Nov 15 2011 *)
Join[{0},Table[FromDigits[PadRight[{},n,1]],{n,20}]] (* Harvey P. Dale, Mar 04 2012 *)

a[0]:0$
a[1]:1$
a[n]:=11*a[n-1]-10*a[n-2]$
A002275(n):=a[n]$
makelist(A002275(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=(10^n-1)/9; \\ Michael B. Porter, Oct 26 2009

my(x='x+O('x^30)); concat(0, Vec(x/((1-10*x)*(1-x)))) \\ Altug Alkan, Apr 10 2016

print([(10**n-1)//9 for n in range(100)]) # Michael S. Branicky, Apr 30 2022

[lucas_number1(n, 11, 10) for n in range(21)]  # Zerinvary Lajos, Apr 27 2009

a(n) = 10*a(n-1) + 1, a(0)=0.
a(n) = A000042(n) for n >= 1.
Second binomial transform of Jacobsthal trisection A001045(3n)/3 (A015565). - Paul Barry, Mar 24 2004
G.f.: x/((1-10*x)*(1-x)). Regarded as base b numbers, g.f. x/((1-b*x)*(1-x)). - Franklin T. Adams-Watters, Jun 15 2006
a(n) = 11*a(n-1) - 10*a(n-2), a(0)=0, a(1)=1. - Lekraj Beedassy, Jun 07 2006
a(n) = A125118(n,9) for n>8. - Reinhard Zumkeller, Nov 21 2006
a(n) = A075412(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 10^(n-1) with a(0)=0. - Vincenzo Librandi, Jul 22 2010
a(n) = A242614(n,A242622(n)). - Reinhard Zumkeller, Jul 17 2014
E.g.f.: (exp(9*x) - 1)*exp(x)/9. - Ilya Gutkovskiy, May 11 2016
a(n) = Sum_{k=0..n-1} 10^k. - Torlach Rush, Nov 03 2020
Sum_{n>=1} 1/a(n) = A065444. - Amiram Eldar, Nov 13 2020
From Elmo R. Oliveira, Aug 02 2025: (Start)
a(n) = A002283(n)/9 = A105279(n)/10.
a(n) = A010785(A017173(n-1)) for n >= 1. (End)

A057951 Number of prime factors of 10^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 3, 4, 4, 4, 7, 4, 6, 6, 6, 4, 9, 5, 6, 8, 8, 4, 11, 3, 9, 9, 9, 3, 12, 7, 8, 9, 10, 7, 15, 5, 13, 8, 8, 9, 14, 5, 5, 8, 13, 6, 17, 6, 13, 12, 8, 4, 15, 6, 12, 10, 11, 6, 16, 10, 14, 8, 10, 4, 22, 9, 7, 16, 17, 9, 17, 5, 12, 8, 14, 4, 20, 5, 9, 14, 8, 10, 18

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Ray Chandler)
Makoto Kamada, Factorizations of 11...11 (Repunit).
S. S. Wagstaff, Jr., Main Tables from the Cunningham Project.
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): this sequence (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), A057956 (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A002283, A046053, A085035, A003020, A046107, A005422, A061075, A102380, A095370, A147556, A081317, A081318, A102347, A112505.

PrimeOmega[10^Range[70]-1] (* Jayanta Basu, May 29 2013 *)

a(n)=bigomega(10^n-1) \\ Charles R Greathouse IV, Sep 14 2015

Mobius transform of A085035 - T. D. Noe, Jun 19 2003
a(n) = Omega(10^n -1) = Omega(R_n) + 2 = A046053(n) + 2 {where Omega(n) = A001222(n) and R_n = (10^n - 1)/9 = A002275(n)}. - Lekraj Beedassy, Jun 09 2006
a(n) = A001222(A002283(n)). - Ray Chandler, Apr 22 2017

Erroneous b-file replaced by Ray Chandler, Apr 26 2017

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

A005422 Largest prime factor of 10^n - 1.

Original entry on oeis.org

3, 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: 1

N. J. A. Sloane

nonn

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Table of n, a(n) for n = 1..352
K. Beschorner, Factorizations of Repunit Numbers of the form (10^n-1)/9 (111111...). [Retrieved July 30, 2015]
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Yousuke Koide, Factorizations of Repunit Numbers.
S. S. Wagstaff, Jr., The Cunningham Project

Same as A003020 except for the additional a(1) = 3.
Cf. A002275, A002283, A004023, A006530, A067063, A075024, A102380.
Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(10^n-1)): n in [1..45]]; // Vincenzo Librandi, Jul 13 2016

A005422 := proc(n)
    10^n-1 ;
    A006530(%) ;
end proc: # R. J. Mathar, Dec 02 2016

Table[FactorInteger[10^n - 1][[-1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Jul 13 2016 *)

a(n)=vecmax(factor(10^n-1)[,1]) \\ Simplified by M. F. Hasler, Jul 30 2015

For n > 1, a(n) = A003020(n). For 1 < n < 10, a(n) = A075024(n). - M. F. Hasler, Jul 30 2015
a(n) = A006530(A002283(n)). - Vincenzo Librandi, Jul 13 2016
a(A004023(n)) = A002275(A004023(n)). - Bernard Schott, May 24 2022

Terms to a(100) in b-file from Yousuke Koide added by T. D. Noe, Dec 06 2006
Edited by M. F. Hasler, Jul 30 2015
a(101)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(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

A070529 Number of divisors of repunit 111...111 (with n digits).

Original entry on oeis.org

1, 2, 4, 4, 4, 32, 4, 16, 12, 16, 4, 128, 8, 16, 64, 64, 4, 384, 2, 128, 128, 96, 2, 1024, 32, 64, 64, 256, 32, 8192, 8, 2048, 64, 64, 128, 3072, 8, 8, 64, 2048, 16, 24576, 16, 1536, 768, 64, 4, 8192, 16, 1024, 256, 512, 16, 8192, 256, 4096

Offset: 1

Henry Bottomley, May 02 2002

			a(9) = 12 since the divisors of 111111111 are 1, 3, 9, 37, 111, 333, 333667, 1001001, 3003003, 12345679, 37037037, 111111111.

Table of n, a(n) for n = 1..352

Cf. A000005, A002275, A070528, A051064, A004023, A046413, A003020, A095370, A176973, A046053, A046412, A102380.

a(n) = numdiv(10^n\9); \\ Michel Marcus, Sep 09 2015

a(n) = A000005(A002275(n)).
a(n) = A070528(n)*A051064(n)/(A051064(n)+2).
a(A004023(n)) = 2. - Michel Marcus, Sep 09 2015
a(A046413(n)) = 4. - Bruno Berselli, Sep 09 2015

Terms to a(280) in b-file from Hans Havermann, Aug 20 2011
a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) ib b-file from Max Alekseyev, May 04 2022

A102146 a(n) = sigma(10^n - 1), where sigma(n) is the sum of positive divisors of n.

Original entry on oeis.org

13, 156, 1520, 15912, 148512, 2042880, 14508000, 162493344, 1534205464, 16203253248, 144451398000, 2063316971520, 14903272088640, 158269280832000, 1614847741624320, 17205180696931968, 144444514193267496

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 14 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..352 (terms 1..322 from Ray Chandler)
C. Caldwell, Sigma function.

Cf. A000203, A001270, A002283, A003020, A005422, A046053, A046107, A046412, A046415, A046416, A046417, A046418, A046419, A046420, A057951, A059892, A061075, A070528, A070529, A081317, A081318, A085035, A095370, A095413, A095414, A095417, A095418, A102347, A102380, A112505, A147556, A295503, A366669.

DivisorSigma[1,10^Range[20]-1] (* Harvey P. Dale, Jan 05 2012 *)

a(n) = sigma(10^n-1); \\ Michel Marcus, Apr 22 2017

a(n) = A000203(A002283(n)). - Ray Chandler, Apr 22 2017

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

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

A204847 Primitive cofactor of n-th repunit A002275(n).

Original entry on oeis.org

1, 11, 111, 101, 11111, 91, 1111111, 10001, 333667, 9091, 11111111111, 9901, 1111111111111, 909091, 90090991, 100000001, 11111111111111111, 999001, 1111111111111111111, 99009901, 900900990991, 826446281, 11111111111111111111111, 99990001, 100001000010000100001

Offset: 1

N. J. A. Sloane, Jan 19 2012

nonn

Except for a(1) = 1 and a(3) = 111, this is the Zsigmondy numbers for a = 10, b = 1: Zs(n, 10, 1) is the greatest divisor of 10^n - 1^n that is coprime to 10^m - 1^m for all positive integers m < n. The prime terms are called unique primes or unique period primes (A007615).

Differs from A019328 for n = 1, 9, 22, 27, 42, ... - Jianing Song, Apr 30 2018

Makoto Kamada, Factorizations of 11...11 (Repunit).
Samuel Yates, Cofactors of repunits, Journal of Recreational Mathematics, Vol. 8(2), pp. 99, 1975-76.
Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Cf. A002275, A019328, A102380, A204845, A204846.

lista(nn) = {vf = []; vfs = []; for (n=1, nn, if (n==1, print1(n, ", "), f = factor((10^n-1)/9)[,1]; vkeep = []; for (k = 1, #f~, if (!vecsearch(vfs, f[k]), vkeep = concat(vkeep, f[k]));); print1(prod(j=1, #vkeep, vkeep[j]), ", "); vf = concat(vf, vkeep); vfs = Set(vf);););} \\ Michel Marcus, May 18 2018

Equals A002275(n)/(product of terms in n-th row of A204845).

a(11)-a(24) from Jianing Song, Apr 30 2018

a(25) from Jinyuan Wang, May 02 2021

cp's OEIS Frontend

A002275 Repunits: (10^n - 1)/9. Often denoted by R_n.

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A057951 Number of prime factors of 10^n - 1 (counted with multiplicity).

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

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A005422 Largest prime factor of 10^n - 1.

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

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A070529 Number of divisors of repunit 111...111 (with n digits).

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