A061075 -id:A061075 - cp's OEIS frontend

A181086 Sorted version of A061075.

3, 11, 13, 37, 101, 137, 271, 2161, 4649, 8779, 9091, 9901, 27961, 52579, 69857, 333667, 459691, 513239, 909091, 2906161, 5882353, 10838689, 39526741, 99990001, 121499449, 265371653, 1056689261, 1058313049, 5363222357, 5964848081

Offset: 1

Jeff Burch, Oct 01 2010

nonn

Warning: There is some doubt as to whether this sequence is correct. It would be good to have confirmation that the terms shown are correct and that there are no missing terms. - Editors of OEIS, May 01 2017

Cf. A061075.

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

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

A007615 Primes with unique period length (the periods are given in A007498).

Original entry on oeis.org

3, 11, 37, 101, 333667, 9091, 9901, 909091, 1111111111111111111, 11111111111111111111111, 99990001, 999999000001, 909090909090909091, 900900900900990990990991, 9999999900000001, 909090909090909090909090909091, 900900900900900900900900900900990990990990990990990990990991

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Additional terms are Phi(n,10)/gcd(n,Phi(n,10)) for the n in A007498, where Phi(n,10) is the n-th cyclotomic polynomial evaluated at 10.

			3 is the only prime p such that decimal expansion of 1/p has (nontrivial) period exactly 1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.

Max Alekseyev, Table of n, a(n) for n = 1..98 (terms 1..25 from T. D. Noe; terms 26..31 from Ray Chandler)
C. K. Caldwell, The Prime Glossary, unique prime
Makoto Kamada, Factorizations of Phi_n(10)
Index entries for sequences related to decimal expansion of 1/n

Cf. A007498, A040017, A002371, A048595, A006883, A007732, A051626, A061075, A006530, A019328.

nmax = 50; periods = Reap[ Do[ p = Cyclotomic[n, 10] / GCD[n, Cyclotomic[n, 10]]; If[ PrimeQ[p], Sow[n]], {n, 1, nmax}]][[2, 1]]; Cyclotomic[#, 10] / GCD[#, Cyclotomic[#, 10]]& /@ periods // Prepend[#, 3]& (* Jean-François Alcover, Mar 28 2013 *)

a(n) = A061075(A007498(n)). - Max Alekseyev, Oct 16 2010

a(n) = A006530(A019328(A007498(n))). - Ray Chandler, May 10 2017

A121341 Number of decimal places before 1/n either recurs or terminates.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 6, 3, 1, 1, 2, 3, 6, 7, 2, 4, 16, 2, 18, 2, 6, 3, 22, 4, 2, 7, 3, 8, 28, 2, 15, 5, 2, 17, 7, 3, 3, 19, 6, 3, 5, 7, 21, 4, 2, 23, 46, 5, 42, 2, 16, 8, 13, 4, 3, 9, 18, 29, 58, 3, 60, 16, 6, 6, 7, 3, 33, 18, 22, 7, 35, 4, 8, 4, 3, 20, 6, 7, 13, 4, 9, 6, 41, 8, 17, 22, 28, 5, 44, 2, 6

Offset: 1

Anthony C Robin, Aug 29 2006

In this sequence, the repeating decimals (e.g., 1/7) are treated differently from nonrepeating decimals (e.g., 1/5). If they are treated the same, then a(2)=2, a(4)=3, a(5)=2, a(8)=4, a(10)=2, ... and we obtain A054710. The two sequence differ only for n = 2^j * 5^k.

			1/592 = 0.0016891891891... starts with 4 decimals (0016, zeros counted) and has period 3 (digits 891) to yield a(592) = 4 + 3 = 7.

T. D. Noe, Table of n, a(n) for n = 1..1000 (corrected by Sean A. Irvine, Apr 29 2022)
Index entries for sequences related to decimal expansion of 1/n

A007732 is the length of the periods and serves as a lower bound. Cf. A061075.

Cf. A051626, A051628.

a[n_] := Max[IntegerExponent[n, 2], IntegerExponent[n, 5]] + Length[RealDigits[1/n][[1, -1]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 20 2022 *)

a(n) = A051628(n) + A051626(n). - Sean A. Irvine, Apr 13 2022

More terms from T. D. Noe, Aug 30 2006

Additional comments from R. J. Mathar, Aug 30 2006

A081317 Primes p such that p divides 10^n-1, p is the largest prime producing decimal fraction period n and p is not the largest prime dividing 10^n-1.

Original entry on oeis.org

13, 52579, 8779, 2161, 69857, 909090909090909091, 459691, 549797184491917, 14175966169, 183411838171, 296557347313446299, 388847808493, 3404193829806058997303, 8985695684401, 297262705009139006771611927

Offset: 1

Hugo Pfoertner, Mar 18 2003

			a(1)=13 because the largest factor 37 in the factorization of 10^6-1=999999=3^3*7*11*13*37 already occurs in the factorization of 10^3-1=3^3*37 and produces only a decimal fraction period of 3. 1/37=0.027027027...., 1/13=0.0769230769230...

Max Alekseyev, Table of n, a(n) for n = 1..54 (first 50 terms from Ray Chandler)
Sam Wagstaff, The Cunningham Project, The Main Tables

Cf. A005422, A003020, A061075, A081318.

Numbers in A061075(n) such that A061075(n) is not equal to A005422(n). The corresponding values of n are given in A081318.

a(n) = A061075(A081318(n)). - Max Alekseyev, Apr 27 2022

More terms from Hans Havermann, May 31 2003

A081318 Integers n such that the reciprocal of the largest prime factor of 10^n-1 is not a repeating decimal fraction with a period of n.

Original entry on oeis.org

6, 18, 22, 30, 32, 38, 42, 46, 54, 66, 74, 78, 82, 90, 94, 96, 110, 118, 132, 138, 146, 154, 162, 174, 186, 194, 198, 206, 210, 218, 228, 231, 240, 242, 254, 258, 260, 264, 266, 268, 274, 282, 284, 286, 298, 300, 306, 310, 318, 322, 334, 338, 344, 348

Offset: 1

Hugo Pfoertner, Mar 18 2003

For all but three of the terms through a(41)=274, the reciprocal of the largest prime factor of 10^a(n)-1 is a decimal fraction with a period of a(n)/2. Of the three exceptions, there are two (a(32)=231 and a(38)=264) where the period is a(n)/3, and one (a(19)=132) where the period is a(n)/4. - Jon E. Schoenfield, Jun 27 2010

			30 is in the sequence because the factorization of 10^30-1 is 3^3*7*11*13*31*37*41*211*241*271*2161*9091*2906161 and 2906161 occurs already in 10^15-1=3^3*31*37*41*271*2906161 producing a decimal fraction with a period of 15, (1/2906161=0.000000344096559000000344096559000000344...)

Makoto Kamada, Factorizations of 11...11 (repunit).

Cf. A081317.

Integers n such that A061075(n) is not equal to A005422(n).

More terms from Hans Havermann, May 31 2003
Terms a(31)-a(37) from Jon E. Schoenfield, Jun 19 2010
Terms a(38)-a(41) added, link added, and earlier comment expanded by Jon E. Schoenfield, Jun 27 2010
a(42)-a(54) from Max Alekseyev, Aug 17 2013, Apr 26 2022

cp's OEIS Frontend

A181086 Sorted version of A061075.

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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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A007615 Primes with unique period length (the periods are given in A007498).

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A121341 Number of decimal places before 1/n either recurs or terminates.

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A081317 Primes p such that p divides 10^n-1, p is the largest prime producing decimal fraction period n and p is not the largest prime dividing 10^n-1.

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A081318 Integers n such that the reciprocal of the largest prime factor of 10^n-1 is not a repeating decimal fraction with a period of n.

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