A112505 -id:A112505 - cp's OEIS frontend

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

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

A046107 Triangle of prime numbers in which n-th row lists all primes p such that 1/p has decimal period n, n >= 1.

Original entry on oeis.org

3, 11, 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

Eric W. Weisstein

The number of numbers in each row n is given by A112505(n).

In the 18th century, the Japanese mathematician Ajima Naonobu (aka Ajima Chokuyen) gave the terms through 5882353 (Smith and Mikami, p. 199). - Jonathan Sondow, May 25 2013

			First rows of irregular triangle are:
       3;
      11;
      37;
     101;
      41,     271;
       7,      13;
     239,    4649;
      73,     137;
  333667;
    9091;
   21649,  513239;
    9901;
      53,      79, 265371653;
  909091;
      31, 2906161;
      17, 5882353;
  ...

Ajima Naonobu (aka Ajima Chokuyen), Fujin Isshũ (Periods of Decimal Fractions).

Rows n=1..352, flattened
David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; chapter X.
Eric Weisstein's World of Mathematics, Decimal Expansion.
Index entries for sequences related to decimal expansion of 1/n

Cf. A007138 (first column), A061075 (last row elements).

pp={}; Do[f=Transpose[FactorInteger[10^n-1]][[1]]; p=Complement[f, pp]; pp=Union[pp, p]; Print[p], {n, 66}] (* T. D. Noe, Sep 08 2005 *)

First 276 rows in b-file from T. D. Noe, Jun 01 2010
Rows n=277..322 in b-file from Ray Chandler, May 01 2017
Rows n=323..352 in b-file from Max Alekseyev, Apr 26 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

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

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

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

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

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A046107 Triangle of prime numbers in which n-th row lists all primes p such that 1/p has decimal period n, n >= 1.

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

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A106305 Divisors of 10^14 - 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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A113116 Divisors of 10^17 - 1.

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