A070528 -id:A070528 - cp's OEIS frontend

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

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

A102347 Number of distinct prime factors of 10^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 3, 5, 3, 5, 3, 7, 4, 5, 6, 7, 3, 8, 2, 8, 7, 7, 2, 10, 6, 7, 5, 9, 6, 13, 4, 12, 6, 7, 8, 11, 4, 4, 6, 12, 5, 14, 5, 11, 9, 7, 3, 13, 5, 11, 8, 10, 5, 12, 9, 13, 6, 9, 3, 20, 8, 6, 13, 16, 8, 14, 4, 11, 6, 13, 3, 17, 4, 8, 12, 7, 9, 15, 7, 16, 10, 8, 4, 21, 8, 9, 10, 15, 6, 21, 13

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 20 2005

nonn

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

Cf. A001221, A002283, A070528.

A102347 := proc(n)
    10^n-1 ;
    A001221(%) ;
end proc: # R. J. Mathar, Dec 02 2016

Table[PrimeNu[10^n-1],{n,100}] (* The program will take a long time to execute *) (* Harvey P. Dale, Jan 18 2015 *)

a(n) = omega(10^n-1); \\ Michel Marcus, Apr 22 2017

a(n) = A001221(A002283(n)) = A001221(10^n - 1).

a(n) = A001221(R_n) + (n^2 mod 3) = A095370(n) + (n^2 mod 3), where R_n = (10^n-1)/9 = A002275(n). That is, a(n) = A095370(n) for n=3k; otherwise a(n) = A095370(n) + 1. - Lekraj Beedassy, Jun 09 2006

Terms to a(280) and a(323)-a(352) in b-file from Max Alekseyev, Dec 28 2011, Apr 26 2022

a(281)-a(322) in b-file from Ray Chandler, Apr 22 2017

A366709 Number of divisors of 12^n-1.

Original entry on oeis.org

2, 4, 4, 16, 4, 32, 8, 64, 16, 16, 12, 256, 8, 64, 64, 512, 8, 512, 4, 192, 32, 48, 16, 4096, 16, 192, 64, 1024, 32, 8192, 32, 2048, 192, 64, 512, 16384, 8, 64, 128, 12288, 16, 12288, 32, 3072, 4096, 256, 8, 262144, 32, 1024, 64, 6144, 128, 65536, 192, 8192

Offset: 1

Sean A. Irvine, Oct 17 2023

nonn

			a(3)=4 because 12^3-1 has divisors {1, 11, 157, 1727}.

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

Cf. A000005, A024140, A046801, A070528, A366707, A366708, A366710, A366683.

a:=n->numtheory[tau](12^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 12^Range[100]-1]
```
PARI
```
a(n) = numdiv(12^n-1);
```

a(n) = sigma0(12^n-1) = A000005(A024140(n)).

A366683 Number of divisors of 11^n-1.

Original entry on oeis.org

4, 16, 16, 40, 12, 192, 16, 96, 32, 96, 16, 1920, 16, 128, 96, 448, 8, 1024, 8, 480, 768, 1024, 32, 18432, 128, 512, 64, 2560, 16, 9216, 32, 2048, 512, 256, 192, 20480, 64, 512, 4096, 4608, 512, 36864, 16, 10240, 384, 2048, 32, 1376256, 128, 4096, 512, 2560

Offset: 1

Sean A. Irvine, Oct 16 2023

nonn

			a(3)=16 because 11^3-1 has divisors {1, 2, 5, 7, 10, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330}.

Max Alekseyev, Table of n, a(n) for n = 1..316

Cf. A024127, A000005, A366681, A366682, A366684, A366685.

Cf. A046801, A070528, A366709.

a:=n->numtheory[tau](11^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 11^Range[100]-1]
```
PARI
```
a(n) = numdiv(11^n-1);
```

a(n) = sigma0(11^n-1) = A000005(A024127(n)).

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

A344897 a(n) is the number of divisors of 10^n + 1.

Original entry on oeis.org

2, 2, 2, 8, 4, 4, 4, 4, 4, 32, 8, 24, 8, 8, 16, 128, 32, 16, 8, 4, 16, 192, 16, 32, 8, 32, 8, 128, 16, 8, 128, 4, 16, 384, 16, 32, 64, 16, 8, 768, 16, 8, 128, 16, 16, 4096, 16, 16, 512, 16, 128, 256, 16, 4, 64, 768, 32, 64, 32, 16, 64, 8, 8, 3072, 8, 64, 256, 4, 16, 1024, 2048, 8, 32, 16, 128, 2048, 64, 3072, 128, 16

Offset: 0

Seiichi Manyama, Jun 01 2021

nonn

a(n) is even because 10^n + 1 is not a square number.

Max Alekseyev, Table of n, a(n) for n = 0..331

Cf. A000005, A000533, A003021, A038371, A046798, A057934, A070528, A119704, A344859, A087021, A087022,

a[0] = 2; a[n_] := DivisorSigma[0, 10^n + 1]; Array[a, 60, 0] (* Amiram Eldar, Jun 01 2021 *)

PARI
```
a(n) = numdiv(10^n+1);
```

a(n) = A000005(A000533(n)).

A366621 Number of divisors of 6^n-1.

Original entry on oeis.org

2, 4, 4, 8, 6, 16, 4, 16, 16, 48, 8, 128, 8, 48, 48, 64, 32, 128, 8, 384, 16, 32, 32, 512, 32, 128, 64, 384, 4, 1536, 8, 512, 64, 256, 96, 8192, 64, 64, 64, 3072, 8, 768, 32, 512, 1536, 256, 16, 8192, 32, 512, 512, 2048, 16, 2048, 96, 12288, 128, 64, 16

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(4)=8 because 6^4-1 has divisors {1, 5, 7, 35, 37, 185, 259, 1295}.

Max Alekseyev, Table of n, a(n) for n = 1..430

Cf. A024062, A000005, A057955, A059888, A274907, A366613, A366620, A366622, A366623.

Cf. A046801, A366575, A366602, A366612, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](6^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 6^Range[100]-1]
```
PARI
```
a(n) = numdiv(6^n-1);
```

a(n) = sigma0(6^n-1) = A000005(A024062(n)).

A366661 Number of divisors of 9^n-1.

Original entry on oeis.org

4, 10, 16, 24, 24, 80, 16, 112, 128, 180, 64, 384, 16, 160, 768, 256, 128, 1280, 64, 864, 768, 640, 32, 14336, 384, 160, 4096, 1536, 256, 23040, 128, 576, 2048, 1280, 768, 12288, 128, 640, 12288, 16128, 128, 61440, 32, 12288, 196608, 320, 512, 131072, 2048

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

			a(2)=10 because 9^2-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

Max Alekseyev, Table of n, a(n) for n = 1..690

Cf. A024101, A000005, A057952, A366660, A366662, A366663.

Cf. A046801, A366575, A366602, A366612, A366621, A366633, A366652, A070528, A366683, A366709.

a:=n->numtheory[tau](9^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 9^Range[100]-1]
```
PARI
```
a(n) = numdiv(9^n-1);
```

a(n) = sigma0(9^n-1) = A000005(A024101(n)).

a(n) = A366575(2*n) = A366575(n) * A366577(n) * (4 + A007814(n)) / (2 * (3 + A007814(n))). - Max Alekseyev, Jan 07 2024

A366602 Number of divisors of 4^n-1.

Original entry on oeis.org

2, 4, 6, 8, 8, 24, 8, 16, 32, 48, 16, 96, 8, 64, 96, 32, 8, 512, 8, 192, 144, 128, 16, 768, 128, 128, 160, 256, 64, 4608, 8, 128, 384, 128, 512, 8192, 32, 128, 192, 768, 32, 9216, 32, 1024, 4096, 512, 64, 6144, 32, 8192, 1536, 1024, 64, 10240, 3072, 2048, 384

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(4)=8 because 4^4-1 has divisors {1, 3, 5, 15, 17, 51, 85, 255}.

Max Alekseyev, Table of n, a(n) for n = 1..1128

Cf. A024036, A000005, A057957, A295501, A366603, A366604.

Cf. A046798, A046801, A366575, A366612, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](4^n-1):
seq(a(n), n=1..100);

DivisorSigma[0,4^Range[100]-1] (* Paolo Xausa, Oct 14 2023 *)

PARI
```
a(n) = numdiv(4^n-1);
```

a(n) = sigma0(4^n-1) = A000005(A024036(n)).

a(n) = A046801(2*n) = A046798(n) * A046801(n). - Max Alekseyev, Jan 07 2024

A366633 Number of divisors of 7^n-1.

Original entry on oeis.org

4, 10, 12, 36, 8, 60, 16, 84, 64, 80, 16, 864, 8, 160, 96, 384, 16, 640, 16, 1536, 96, 160, 32, 16128, 32, 80, 1280, 1152, 32, 3840, 32, 1728, 384, 80, 128, 18432, 32, 160, 192, 14336, 32, 7680, 16, 4608, 2048, 160, 16, 147456, 256, 640, 768, 1152, 32, 25600

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(5)=8 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 168061}.

Max Alekseyev, Table of n, a(n) for n = 1..388

Cf. A024075, A000005, A057954, A059889, A074249, A218358, A366632, A366634, A366635.

Cf. A046801, A366575, A366602, A366612, A366621, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](7^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 7^Range[100]-1]
```
PARI
```
a(n) = numdiv(7^n-1);
```

a(n) = sigma0(7^n-1) = A000005(A024075(n)).

cp's OEIS Frontend

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

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A102347 Number of distinct prime factors of 10^n - 1.

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A366709 Number of divisors of 12^n-1.

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A366683 Number of divisors of 11^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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A344897 a(n) is the number of divisors of 10^n + 1.

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A366621 Number of divisors of 6^n-1.

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