A007138 -id:A007138 - cp's OEIS frontend

A002283 a(n) = 10^n - 1.

0, 9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999, 9999999999, 99999999999, 999999999999, 9999999999999, 99999999999999, 999999999999999, 9999999999999999, 99999999999999999, 999999999999999999, 9999999999999999999, 99999999999999999999, 999999999999999999999, 9999999999999999999999

Offset: 0

N. J. A. Sloane

A friend from Germany remarks that the sequence 9, 99, 999, 9999, 99999, 999999, ... might be called the grumpy German sequence: nein!, nein! nein!, nein! nein! nein!, ...
The Regan link shows that integers of the form 10^n -1 have binary representations with exactly n trailing 1 bits. Also those integers have quinary expressions with exactly n trailing 4's. For example, 10^4 -1 = (304444)5. The first digits in quinary correspond to the number 2^n -1, in our example (30)5 = 2^4 -1. A similar pattern occurs in the binary case. Consider 9 = (1001)2. - Washington Bomfim Dec 23 2010
a(n) is the number of positive integers with less than n+1 digits. - Bui Quang Tuan, Mar 09 2015
From Peter Bala, Sep 27 2015: (Start)
For n >= 1, the simple continued fraction expansion of sqrt(a(2*n)) = [10^n - 1; 1, 2*(10^n - 1), 1, 2*(10^n - 1), ...] has period 2. The simple continued fraction expansion of sqrt(a(2*n))/a(n) = [1; 10^n - 1, 2,  10^n - 1, 2, ...] also has period 2. Note the occurrence of large partial quotients in both expansions.
A theorem of Kuzmin in the measure theory of continued fractions says that large partial quotients are the exception in continued fraction expansions.
Empirically, we also see the presence of unexpectedly large partial quotients early in the continued fraction expansions of the m-th roots of the numbers a(m*n) for m >= 3. Some typical examples are given below. (End)
For n > 0, numbers whose smallest decimal digit is 9. - Stefano Spezia, Nov 16 2023

			From _Peter Bala_, Sep 27 2015: (Start)
Continued fraction expansions showing large partial quotients:
a(12)^(1/3) = [9999; 1, 299999998, 1, 9998, 1, 449999998, 1, 7998, 1, 535714284, 1, 2, 2, 142, 2, 2, 1, 599999999, 3, 1, 1,...].
Compare with a(30)^(1/3) = [9999999999; 1, 299999999999999999998, 1, 9999999998, 1, 449999999999999999998, 1, 7999999998, 1, 535714285714285714284, 1, 2, 2, 142857142, 2, 2, 1, 599999999999999999999, 3, 1, 1,...].
a(24)^(1/4) = [999999; 1, 3999999999999999998, 1, 666665, 1, 1, 1, 799999999999999999, 3, 476190, 7, 190476190476190476, 21, 43289, 1, 229, 1, 1864801864801863, 1, 4, 6,...].
Compare with a(48)^(1/4) = [999999999999; 1, 3999999999999999999999999999999999998, 1, 666666666665, 1, 1, 1, 799999999999999999999999999999999999, 3, 476190476190, 7, 190476190476190476190476190476190476, 21, 43290043289, 1, 229, 1, 1864801864801864801864801864801863, 1, 4, 6,...].
a(25)^(1/5) = [99999, 1, 499999999999999999998, 1, 49998, 1, 999999999999999999998, 1, 33332, 3, 151515151515151515151, 5, 1, 1, 1947, 1, 1, 38, 3787878787878787878, 1, 3, 5,...].
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Peter Bala, A002283 and some continued fraction expansions.
Rick Regan, Nines in quinary, September 8th, 2009.
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.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A000533, A003020, A007138, A007953, A008591, A010888, A048379, A066138, A073668, A168624, A276352.
Cf. A002275, A002276, A002277, A002278, A002279, A002280, A002281, A002282.
Cf. A075412, A075415, A178630, A178631, A178632, A178633, A178634, A178635.
Cf. A010785.

a002283 = subtract 1 . (10 ^)  -- Reinhard Zumkeller, Feb 21 2014

[(10^n-1): n in [0..20]]; // Vincenzo Librandi, Apr 26 2011

Table[10^n - 1, {n, 0, 22}] (* Michael De Vlieger, Sep 27 2015 *)

A002283(n):=10^n-1$
makelist(A002283(n),n,0,20); /* Martin Ettl, Nov 08 2012 */

a(n)=10^n-1; \\ Charles R Greathouse IV, Jan 30 2012

def a(n): return 10**n-1 # Michael S. Branicky, Feb 25 2023

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-10*x)-1/(1-x).
E.g.f.: e^(10*x)-e^x. (End)
a(n) = A075412(n)/A002275(n) = A178630(n)/A002276(n) = A178631(n)/A002277(n) = A075415(n)/A002278(n) = A178632(n)/A002279(n) = A178633(n)/A002280(n) = A178634(n)/A002281(n) = A178635(n)/A002282(n). - Reinhard Zumkeller, May 31 2010
a(n) = a(n-1) + 9*10^(n-1) with a(0)=0; Also: a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=9. - Vincenzo Librandi, Jul 22 2010
For n>0, A007953(a(n)) = A008591(n) and A010888(a(n)) = 9. - Reinhard Zumkeller, Aug 06 2010
A048379(a(n)) = 0. - Reinhard Zumkeller, Feb 21 2014
a(n) = Sum_{k=1..n} 9*10^k. - Carauleanu Marc, Sep 03 2016
Sum_{n>=1} 1/a(n) = A073668. - Amiram Eldar, Nov 13 2020
From Elmo R. Oliveira, Jul 19 2025: (Start)
a(n) = 9*A002275(n).
a(n) = A010785(A008591(n)). (End)

More terms from Michael De Vlieger, Sep 27 2015

A112927 a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists.

Original entry on oeis.org

1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 47, 241, 601, 2731, 262657, 29, 233, 331, 2147483647, 65537, 599479, 43691, 71, 37, 223, 174763, 79, 61681, 13367, 5419, 431, 397, 631, 2796203, 2351, 97, 4432676798593, 251, 103, 53, 6361, 87211

Offset: 1

Vladimir Shevelev, Aug 25 2008

nonn

If a(n) differs from 1, then a(n) is the minimal prime divisor of A064078(n);
a(n)=n+1 iff n+1 is prime from A001122; a(n)=2n+1 iff 2n+1 is prime from A115591.
If a(n) > 1 then a(n) is the index where n occurs first in A014664. - M. F. Hasler, Feb 21 2016
Bang's theorem (special case of Zsigmondy's theorem, see links): a(n)>1 for all n>6. - Jeppe Stig Nielsen, Aug 31 2020

Robert G. Wilson v, Table of n, a(n) for n = 1..1206
Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
Will Edgington, Factored Mersenne Numbers [from Internet Archive Wayback Machine]
Wikipedia, Zsigmondy's theorem

Cf. A002326, A064078, A001122, A115591.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

A112927(n,f=factor(2^n-1)[,1])=!for(i=1,#f,znorder(Mod(2,f[i]))==n&&return(f[i])) \\ Use the optional 2nd arg to give a list of pseudoprimes to try when factoring of 2^n-1 is too slow. You may try factor(2^n-1,0)[,1]. - M. F. Hasler, Feb 21 2016

A059892 a(n) = |{m : multiplicative order of 10 mod m is equal to n}|.

Original entry on oeis.org

3, 3, 5, 6, 9, 53, 9, 36, 12, 33, 9, 186, 21, 33, 111, 144, 9, 564, 3, 330, 239, 273, 3, 1756, 84, 165, 76, 714, 93, 16167, 21, 5952, 111, 177, 363, 4288, 21, 15, 99, 5724, 45, 48807, 45, 4314, 1140, 183, 9, 14192, 36, 2940, 495, 1338, 45, 11572, 747, 11484

Offset: 1

Vladeta Jovovic, Feb 06 2001

The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).
The number of unit fractions 1/k having a decimal expansion of period n and with k coprime to 10. - T. D. Noe, May 18 2007
Also, number of primitive factors of 10^n - 1 (cf. A003060). - Max Alekseyev, May 03 2022
a(n) is odd if and only if n is squarefree. Proof: Note that 10^d - 1 == 3 (mod 4) for d >= 2, so 10^d - 1 is a square if and only if d = 1. From the formula we can see that a(n) is odd if and only if mu(n) is nonzero, or n is squarefree. - Jianing Song, Jun 15 2021

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

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), this sequence (b=10).
Cf. A000005, A008683, A002329, A007138, A005422, A057951, A058946, A070528.
Column k=10 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(10^d-1), d=divisors(n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 12 2012

f[n_, d_] := MoebiusMu[n/d]*Length[Divisors[10^d - 1]]; a[n_] := Total[(f[n, #] & ) /@ Divisors[n]]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Mar 21 2011 *)

j=[]; for(n=1,10,j=concat(j,sumdiv(n,d,moebius(n/d)*numdiv(10^d-1)))); j

from sympy import divisors, mobius, divisor_count
def a(n): return sum(mobius(n//d)*divisor_count(10**d - 1) for d in divisors(n)) # Indranil Ghosh, Apr 23 2017

a(n) = Sum_{d|n} mu(n/d)*tau(10^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

More terms from Jason Earls, Aug 06 2001.
Terms to a(280) in b-file from T. D. Noe, Oct 01 2013
a(281)-a(322) in b-file from Ray Chandler, May 03 2017
a(323)-a(352) in b-file from Max Alekseyev, May 03 2022

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

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

Original entry on oeis.org

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

A143665 a(n) is the least prime such that the multiplicative order of 5 mod a(n) equals n.

Original entry on oeis.org

2, 3, 31, 13, 11, 7, 19531, 313, 19, 521, 12207031, 601, 305175781, 29, 181, 17, 409, 5167, 191, 41, 379, 23, 8971, 390001, 101, 5227, 109, 234750601, 59, 61, 1861, 2593, 199, 3061, 211, 37, 149, 761, 79, 241, 2238236249, 43, 1644512641, 89, 1171, 47

Offset: 1

Vladimir Shevelev, Aug 28 2008

nonn

Max Alekseyev, Table of n, a(n) for n = 1..520

Cf. A064081, A218357.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

p = 2; t = Table[0, {100}]; While[p < 3000000001, a = MultiplicativeOrder[5, p]; If[0 < a < 101 && t[[a]] == 0, t[[a]] = p]; p = NextPrime@ p]; t (* Robert G. Wilson v, Oct 13 2014 *)

a(23)-a(40) from Robert G. Wilson v, Oct 13 2014

a(41)-a(46) from Robert G. Wilson v, Oct 15 2014

A143663 a(n) is the least prime such that the multiplicative order of 3 mod a(n) equals n, or a(n)=1 if no such prime exists.

Original entry on oeis.org

2, 1, 13, 5, 11, 7, 1093, 41, 757, 61, 23, 73, 797161, 547, 4561, 17, 1871, 19, 1597, 1181, 368089, 67, 47, 6481, 8951, 398581, 109, 29, 59, 31, 683, 21523361, 2413941289, 103, 71, 530713, 13097927, 2851, 313, 42521761, 83, 43, 431, 5501, 181, 23535794707

Offset: 1

Vladimir Shevelev, Aug 28 2008

nonn

If a(n) differs from 1, then a(n) is the minimal prime divisor of A064079(n).

Max Alekseyev, Table of n, a(n) for n = 1..730 (first 153 terms from Robert G. Wilson v)

Cf. A002326, A064079, A112092, A218356, A057958, A074477.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

a:= proc(n) local f,p;
f:= numtheory:-factorset(3^n - 1);
for  p in f do
   if numtheory:-order(3,p) = n then return p fi
od:
1
end proc:
seq(a(n),n=1..100); # Robert Israel, Oct 13 2014

p = 2; t = Table[0, {100}]; While[p < 100000001, a = MultiplicativeOrder[3, p]; If[0 < a < 101 && t[[a]] == 0, t[[a]] = p; Print[{a, p}]];  p = NextPrime@ p]; t (* Robert G. Wilson v, Oct 13 2014 *)

More terms from Robert G. Wilson v, Dec 11 2013

A085035 Number of prime factors of cyclotomic(n,10), which is A019328(n), the value of the n-th cyclotomic polynomial evaluated at x=10.

Original entry on oeis.org

2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 3, 4, 1, 1, 3, 2, 3, 3, 5, 3, 3, 5, 2, 3, 3, 1, 3, 1, 1, 2, 4, 4, 4, 3, 2, 4, 2, 1, 2, 3, 4, 2, 4, 2, 4, 2, 3, 2, 2, 3, 7, 1, 5, 4, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 4, 5, 6, 2, 6, 2, 3, 2, 3, 3, 3

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057951, number of prime factors of 10^n-1.

See references at A085021.

Max Alekseyev, Table of n, a(n) for n = 1..352 (first 322 terms from Ray Chandler)
Makoto Kamada, Factorizations of Phi_n(10)

omega(Phi(n,x)): A085021 (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), this sequence (x=10).

Cf. A001222, A003060, A005422, A007138, A019328, A057951, A070528, A059892.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 10]]][[2]], {n, 1, 100}]

a(n) = A001222(A019328(n)). - Ray Chandler, May 10 2017

A112092 a(n) is the least prime such that the multiplicative order of 4 mod a(n) equals n.

Original entry on oeis.org

3, 5, 7, 17, 11, 13, 43, 257, 19, 41, 23, 241, 2731, 29, 151, 65537, 43691, 37, 174763, 61681, 337, 397, 47, 97, 251, 53, 87211, 15790321, 59, 61, 715827883, 641, 67, 137, 71, 433, 223, 229, 79, 4278255361, 83, 1429, 431, 353, 631, 277, 283, 193, 4363953127297

Offset: 1

Vladimir Shevelev, Aug 28 2008

nonn

a(n) is the minimal prime divisor of A064080(n).

Max Alekseyev, Table of n, a(n) for n = 1..1207 (first 100 terms from Amiram Eldar)

Cf. A064080, A112927.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

a[n_] := Module[{f = FactorInteger[4^n - 1][[;; , 1]]}, Do[p = f[[k]]; If[ MultiplicativeOrder[4, p] == n, Break[] ], {k, 1, Length[f]}]; p]; Array[a, 100] (* Amiram Eldar, Jan 27 2019 *)

a(n) = {my(p = 3); while (znorder(Mod(4, p)) != n, p = nextprime(p+1)); p;} \\ Michel Marcus, Feb 08 2016

a(29)-a(30) from Michel Marcus, Feb 08 2016

More term from Amiram Eldar, Jan 27 2019

A252170 Smallest primitive prime factor of 12^n-1.

Original entry on oeis.org

11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, 3933841, 3307

Offset: 1

Eric Chen, Dec 15 2014

nonn

Also, smallest prime p such that 1/p has duodecimal period n.

			a(4) = 5 because 1/5 = 0.249724972497... and 5 is the smallest prime with period 4 in base 12.
a(5) = 22621 because 1/22621 = 0.0000100001... and 22621 is the smallest (in fact, the only one) prime with period 5 in base 12.

Max Alekseyev, Table of n, a(n) for n = 1..310

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

Cf. A246004, A246489.

S:= {}:
for n from 1 to 72 do
  F:= numtheory:-factorset(12^n-1) minus S;
  A[n]:= min(F);
  S:= S union F;
od:
seq(A[n], n=1..72);

prms={}; Table[f=First/@FactorInteger[12^n-1]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, First[p]], {n, 72}]

listap(nn) = {prf = []; for (n=1, nn, vp = (factor(12^n-1)[,1])~; f = setminus(Set(vp), Set(prf)); prf = concat(prf, f); print1(vecmin(Vec(f)), ", "););} \\ Michel Marcus, Dec 15 2014; after A007138

Edited by Max Alekseyev, Aug 26 2021

cp's OEIS Frontend

A002283 a(n) = 10^n - 1.

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A112927 a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists.

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A059892 a(n) = |{m : multiplicative order of 10 mod m is equal to n}|.

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

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A143665 a(n) is the least prime such that the multiplicative order of 5 mod a(n) equals n.

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A143663 a(n) is the least prime such that the multiplicative order of 3 mod a(n) equals n, or a(n)=1 if no such prime exists.

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