A027889 -id:A027889 - cp's OEIS frontend

A070528 Number of divisors of 10^n-1 (999...999 with n digits).

3, 6, 8, 12, 12, 64, 12, 48, 20, 48, 12, 256, 24, 48, 128, 192, 12, 640, 6, 384, 256, 288, 6, 2048, 96, 192, 96, 768, 96, 16384, 24, 6144, 128, 192, 384, 5120, 24, 24, 128, 6144, 48, 49152, 48, 4608, 1280, 192, 12, 16384, 48, 3072, 512, 1536, 48, 12288, 768

Offset: 1

Henry Bottomley, May 02 2002

			a(7)=12 since the divisors of 9999999 are 1, 3, 9, 239, 717, 2151, 4649, 13947, 41841, 1111111, 3333333, 9999999.

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

Cf. A004023 (indices of 6's).
Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895.
Cf. A046801, A005422, A102347, A046053, A057951, A007138, A003060, A059892, A085035.

DivisorSigma[0,#]&/@(10^Range[60]-1) (* Harvey P. Dale, Jan 14 2011 *)
Table[DivisorSigma[0, 10^n - 1], {n, 60}] (* T. D. Noe, Aug 18 2011 *)

a(n) = numdiv(10^n - 1); \\ Michel Marcus, Sep 08 2015

a(n) = A000005(A002283(n)).
a(n) = Sum_{d|n} A059892(d).
a(n) = A070529(n)*(A007949(n)+3)/(A007949(n)+1).

Terms to a(280) in b-file from Hans Havermann, Aug 19 2011
a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) in b-file from Max Alekseyev, May 04 2022

A066364 Prime divisors of solutions to 10^n == 1 (mod n).

Original entry on oeis.org

3, 37, 163, 757, 1999, 5477, 8803, 9397, 13627, 15649, 36187, 40879, 62597, 106277, 147853, 161839, 215893, 231643, 281683, 295759, 313471, 333667, 338293, 478243, 490573, 607837, 647357, 743933, 988643, 1014877, 1056241, 1168711, 1353173, 1390757, 1487867, 1519591, 1627523, 1835083, 1912969, 2028119, 2029759, 2064529

Offset: 1

Vladeta Jovovic, Dec 21 2001

nonn

			10^27-1 = 3^5*37*757*333667*440334654777631 is a solution to the congruence.

Max Alekseyev and Hans Havermann (Max Alekseyev to 501), Table of n, a(n) for n = 1..2060
Rüdiger Jehn and Kester Habermann, Properties of terms of OEIS A342810, arXiv:2106.05866 [math.GM], 2021.
Makoto Kamada, Factorizations of 11...11 (Repunit).

Cf. A014950, A001270, A027889, A007138, A114207.

fQ[p_] := Block[{fi = First@# & /@ FactorInteger[ MultiplicativeOrder[ 10, p]]}, Union[ MemberQ[ lst, #] & /@ fi] == {True}]; k = 4; lst = {3}; While[k < 180000, If[ p = Prime@ k; fQ@ p, AppendTo[ lst, p]; Print@ p]; k++]; lst (* Robert G. Wilson v, Nov 30 2013 *)

S=Set([3]); forprime(p=7,10^6, v=factorint(znorder(Mod(10,p)))[,1]; if(length(setintersect(S,Set(v)))==length(v), S=setunion(S,[p])) ); print(vecsort(eval(S))) \\ Max Alekseyev, Nov 16 2005

A prime p is a term iff all prime divisors of ord_p(10) are terms, where ord_p(10) is the order of 10 modulo p. - Max Alekseyev, Nov 16 2005

Edited by Max Alekseyev, Nov 16 2005

Edited by Hans Havermann, Jul 11 2014

A106305 Divisors of 10^14 - 1.

Original entry on oeis.org

1, 3, 9, 11, 33, 99, 239, 717, 2151, 2629, 4649, 7887, 13947, 23661, 41841, 51139, 153417, 460251, 909091, 1111111, 2727273, 3333333, 8181819, 9999999, 10000001, 12222221, 30000003, 36666663, 90000009, 109999989, 217272749, 651818247

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 12 2005

Nathaniel Johnston, Table of n, a(n) for n = 1..48 (full sequence)
Index entries for sequences related to divisors of numbers

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A109933, A001270, A007138, A046107, A112505.

Divisors[10^14 - 1] (* Harvey P. Dale, Feb 01 2015 *)

divisors(10^14-1) \\ Charles R Greathouse IV, Jun 21 2017

10^14 - 1 = 3^2 * 11 * 239 * 4649 * 909091 = 99999999999999. - Alonso del Arte, Nov 09 2017

A109492 Divisors of 111111.

Original entry on oeis.org

1, 3, 7, 11, 13, 21, 33, 37, 39, 77, 91, 111, 143, 231, 259, 273, 407, 429, 481, 777, 1001, 1221, 1443, 2849, 3003, 3367, 5291, 8547, 10101, 15873, 37037, 111111

Offset: 1

Philippe Deléham, Aug 28 2005

Note that the smaller repunits R3=111, R4=1111, R5=11111 are semiprimes and have only 4 divisors, which is again the case for the next repunit R7=1111111. - M. F. Hasler, Oct 13 2011

Eric Weisstein's World of Mathematics, Repunit
Index entries for sequences related to divisors of numbers

Cf. A070529, A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895. [M. F. Hasler, Oct 13 2011]

Cf. A199799 (totatives of 111111), A154549 (111111*n).

Divisors[111111] (* Paolo Xausa, Sep 27 2023 *)

divisors(111111) \\ Michel Marcus, Sep 08 2015

A109933 Divisors of 10^13 - 1.

Original entry on oeis.org

1, 3, 9, 53, 79, 159, 237, 477, 711, 4187, 12561, 37683, 265371653, 796114959, 2388344877, 14064697609, 20964360587, 42194092827, 62893081761, 126582278481, 188679245283, 1111111111111, 3333333333333, 9999999999999

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 11 2005

Index entries for sequences related to divisors of numbers

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A001270, A007138, A046107.

Divisors[10^13 - 1] (* Wesley Ivan Hurt, Feb 14 2014 *)

divisors(10^13-1) \\ Charles R Greathouse IV, Jun 21 2017

A111117 Divisors of 10^15 - 1.

Original entry on oeis.org

1, 3, 9, 27, 31, 37, 41, 93, 111, 123, 271, 279, 333, 369, 813, 837, 999, 1107, 1147, 1271, 1517, 2439, 3441, 3813, 4551, 7317, 8401, 10027, 10323, 11111, 11439, 13653, 25203, 30081, 30969, 33333, 34317, 40959, 47027, 75609, 90243, 99999, 141081

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 15 2005

Harry J. Smith, Table of n, a(n) for n = 1..128
Index entries for sequences related to divisors of numbers

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A109933, A106305, A001270, A007138, A046107, A112505.

Divisors[10^15-1] (* Harvey P. Dale, Jul 22 2014 *)

{ divisors(10^15-1) } \\ Harry J. Smith, Mar 08 2009

A111211 Divisors of 10^16 - 1.

Original entry on oeis.org

1, 3, 9, 11, 17, 33, 51, 73, 99, 101, 137, 153, 187, 219, 303, 411, 561, 657, 803, 909, 1111, 1233, 1241, 1507, 1683, 1717, 2329, 2409, 3333, 3723, 4521, 5151, 6987, 7227, 7373, 9999, 10001, 11169, 13563, 13651, 13837, 15453, 18887, 20961, 22119, 25619

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 25 2005

Nathaniel Johnston, Table of n, a(n) for n = 1..192 (full sequence)
Index entries for sequences related to divisors of numbers

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A109933, A106305, A111117, A001270, A007138, A046107, A112505.

Divisors[10^16-1] (* Harvey P. Dale, Aug 30 2014 *)

divisors(10^16-1) \\ Charles R Greathouse IV, Jun 21 2017

A113522 Divisors of 10^18 - 1.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 19, 21, 27, 33, 37, 39, 57, 63, 77, 81, 91, 99, 111, 117, 133, 143, 171, 189, 209, 231, 247, 259, 273, 297, 333, 351, 399, 407, 429, 481, 513, 567, 627, 693, 703, 741, 777, 819, 891, 999, 1001, 1053, 1197, 1221, 1287, 1443, 1463, 1539, 1729

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jan 12 2006

Nathaniel Johnston, Table of n, a(n) for n = 1..640 (full sequence)
Index entries for sequences related to divisors of numbers

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A109933, A106305, A111117, A111211, A113116, A001270, A007138, A046107, A112505.

Divisors[10^18 - 1][[;;100]] (* Paolo Xausa, Jul 31 2024 *)

divisors(10^18-1) \\ Charles R Greathouse IV, Jun 21 2017

A070189 a(n) = 12345679*n.

Original entry on oeis.org

0, 12345679, 24691358, 37037037, 49382716, 61728395, 74074074, 86419753, 98765432, 111111111, 123456790, 135802469, 148148148, 160493827, 172839506, 185185185, 197530864, 209876543, 222222222, 234567901, 246913580, 259259259, 271604938, 283950617, 296296296, 308641975

Offset: 0

Henry Bottomley, Apr 24 2002

a(82)=1012345678 is the first term which has a digit appearing more than once without an obvious pattern, although a(-82)=-1012345678 might be seen as the concatenation of ten consecutive numbers.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 12345679 at p. 188.

Tanya Khovanova, Recursive Sequences.
"TyYann", MegaFavNumbers: Lewis Carroll's Number, video (2020).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A027889, A057932, A060073, A085959.

Table[12345679*n,{n,0,30}] (* or *) LinearRecurrence[{2,-1},{0,12345679},30] (* Harvey P. Dale, Oct 16 2015 *)

a(n)=12345679*n \\ Charles R Greathouse IV, Jan 09 2012

concat(0, Vec(12345679*x/(1-x)^2 + O(x^26))) \\ Elmo R. Oliveira, Jun 26 2025

a(n) = n*(10^(10-1)-1)/(10-1)^2.
From Elmo R. Oliveira, Jun 26 2025: (Start)
G.f.: 12345679*x/(1-x)^2.
E.g.f.: 12345679*x*exp(x).
a(n) = 333667*A085959(n).
a(n) = 2*a(n-1) - a(n-2). (End)

More terms from Elmo R. Oliveira, Jun 26 2025

A113116 Divisors of 10^17 - 1.

Original entry on oeis.org

1, 3, 9, 2071723, 6215169, 18645507, 5363222357, 16089667071, 48269001213, 11111111111111111, 33333333333333333, 99999999999999999

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jan 02 2006

Index entries for sequences related to divisors of numbers

Cf. A018282, A018766, A027894, A027893, A027892, A027891, A027890, A027889, A027895, A027896, A027897, A109933, A106305, A111117, A111211, A001270, A007138, A046107, A112505.

Divisors[10^17-1] (* Harvey P. Dale, Jan 20 2013 *)

divisors(10^17-1) \\ Charles R Greathouse IV, Jun 21 2017

cp's OEIS Frontend

A070528 Number of divisors of 10^n-1 (999...999 with n digits).

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A066364 Prime divisors of solutions to 10^n == 1 (mod n).

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A106305 Divisors of 10^14 - 1.

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A109492 Divisors of 111111.

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A109933 Divisors of 10^13 - 1.

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A111117 Divisors of 10^15 - 1.

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A111211 Divisors of 10^16 - 1.

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A113522 Divisors of 10^18 - 1.

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