A005422 -id:A005422 - 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

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

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

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

A274906 Largest prime factor of 4^n - 1.

Original entry on oeis.org

3, 5, 7, 17, 31, 13, 127, 257, 73, 41, 683, 241, 8191, 127, 331, 65537, 131071, 109, 524287, 61681, 5419, 2113, 2796203, 673, 4051, 8191, 262657, 15790321, 3033169, 1321, 2147483647, 6700417, 599479, 131071, 122921, 38737, 616318177, 525313, 22366891

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			4^7 - 1 = 16383 = 3*43*127, so a(7) = 127

Max Alekseyev, Table of n, a(n) for n = 1..1128
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Second bisection of A005420. - Michel Marcus, Jul 13 2016
Cf. largest prime factor of k^n-1: A005420 (k=2), A074477 (k=3), this sequence (k=4), A074479 (k=5), A274907 (k=6), A074249 (k=7), A274908 (k=8), A274909 (k=9), A005422 (k=10), A274910 (k=11).
Cf. A006530, A024036.

[Maximum(PrimeDivisors(4^n-1)): n in [1..40]];

Table[FactorInteger[4^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024036(n)). - Michel Marcus, Jul 11 2016

a(n) = max(A002587(n),A005420(n)). - Max Alekseyev, Apr 25 2022

Terms to a(100) in b-file from Vincenzo Librandi, Jul 13 2016
a(101)-a(603) in b-file from Amiram Eldar, Feb 08 2020
a(604)-a(1128) in b-file from Max Alekseyev, Jul 25 2023, Mar 15 2025

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

A003060 Smallest number with reciprocal of period length n in decimal (base 10).

Original entry on oeis.org

1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, 21649, 707, 53, 2629, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 511, 21401, 583, 243, 29, 3191, 211, 2791, 353, 67, 103, 71, 1919, 2028119, 909090909090909091

Offset: 0

N. J. A. Sloane

For n > 0, a(n) is the least divisor d > 1 of 10^n - 1 such that the multiplicative order of 10 mod d is n. For prime n > 3, a(n) = A007138(n). - T. D. Noe, Aug 07 2007

For n > 1, a(n) is the smallest positive d such that d divides 10^n - 1 and does not divide any of 10^k - 1 for 0 < k < n. - Maciej Ireneusz Wilczynski, Sep 06 2012, corrected by M. F. Hasler, Jun 28 2022. (For n = 1, d = 1 divides 10^n - 1 and does not divide any 10^k - 1 with 0 < k < n, but a(1) = 3 > 1.)

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
"Cycle lengths of reciprocals", Popular Computing (Calabasas, CA), Vol. 1 (No. 4, Jul 1973), pp. 12-14.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Smallest primitive divisors of b^n-1: A212953 (b=2), A218356 (b=3), A218357 (b=5), A218358 (b=7), this sequence (b=10), A218359 (b=11), A218360 (b=13), A218361 (b=17), A218362 (b=19), A218363 (b=23), A218364 (b=29).

Cf. A007138, A057951, A005422, A176973.

a[n_] := First[ Select[ Divisors[10^n - 1], MultiplicativeOrder[10, #] == n &, 1]]; a[0] = 1; a[1] = 3; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jul 13 2012, after T. D. Noe *)

apply( {A003060(n)=!fordiv(10^n-!!n, d, d>1 && znorder(Mod(10,d))==n && return(d))}, [0..50]) \\ M. F. Hasler, Jun 28 2022

Comment corrected by T. D. Noe, Apr 15 2010
More terms from T. D. Noe, Apr 15 2010
b-file truncated at uncertain term a(439) by Max Alekseyev, Apr 30 2022

A061075 Greatest prime number p(n) with decimal fraction period of length n.

Original entry on oeis.org

3, 11, 37, 101, 271, 13, 4649, 137, 333667, 9091, 513239, 9901, 265371653, 909091, 2906161, 5882353, 5363222357, 52579, 1111111111111111111, 27961, 10838689, 8779, 11111111111111111111111, 99990001, 182521213001, 1058313049, 440334654777631, 121499449, 77843839397

Offset: 1

Heiner Muller-Merbach (hmm(AT)sozwi.uni-kl.de), May 29 2001

			1/271 = 0.0036900369, period of n=5 for p(5)=271.

Cf. A003020, A005422, A006530, A007138, A019328.

Last terms in rows of A046107.

a[n_] := Cyclotomic[n, 10] // FactorInteger // Last // First; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Aug 05 2013, after Pari *)

a(n) = my(p); if(n<1, 0, p=factor(polcyclo(n,10))[,1]; p[#p])

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

Terms to a(322) in b-file from Ray Chandler, Apr 28 2017

a(323)-a(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

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

cp's OEIS Frontend

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

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

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

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

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A274906 Largest prime factor of 4^n - 1.

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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.

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A003060 Smallest number with reciprocal of period length n in decimal (base 10).

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