A059890 -id:A059890 - 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

A212957 A(n,k) is the number of moduli m such that the multiplicative order of k mod m equals n; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 3, 2, 2, 2, 0, 2, 5, 4, 6, 1, 0, 4, 2, 3, 4, 4, 3, 0, 2, 6, 2, 12, 6, 10, 1, 0, 4, 4, 8, 4, 9, 16, 2, 4, 0, 3, 6, 2, 26, 4, 37, 6, 14, 2, 0, 4, 3, 12, 18, 4, 10, 3, 8, 4, 5, 0, 2, 12, 5, 14, 6, 42, 2, 28, 26, 16, 3, 0

Offset: 1

Alois P. Heinz, Jun 01 2012

			A(4,3) = 6: 3^4 = 81 == 1 (mod m) for m in {5,10,16,20,40,80}.
Square array A(n,k) begins:
  0,  1,  2,  2,  3,  2,  4,  2, ...
  0,  1,  2,  2,  5,  2,  6,  4, ...
  0,  1,  2,  4,  3,  2,  8,  2, ...
  0,  2,  6,  4, 12,  4, 26, 18, ...
  0,  1,  4,  6,  9,  4,  4,  6, ...
  0,  3, 10, 16, 37, 10, 42, 24, ...
  0,  1,  2,  6,  3,  2, 12, 10, ...
  0,  4, 14,  8, 28,  8, 48, 72, ...

Alois P. Heinz, Antidiagonals n = 1..60
Wikipedia, Multiplicative order.

Columns k=1-10 give: A000004, A059499, A059885, A059886, A059887, A059888, A059889, A059890, A059891, A059892.
Rows n=1-10 give: A000005, A059907, A059908, A059909, A059910, A059911, A218256, A218257, A218258, A218259.
Main diagonal gives A252760.
Cf. A000005, A008683.

with(numtheory):
A:= (n, k)-> add(mobius(n/d)*tau(k^d-1), d=divisors(n)):
seq(seq(A(n, 1+d-n), n=1..d), d=1..15);

a[n_, k_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, k^d - 1], {d, Divisors[n]}]; a[1, 1] = 0; Table[ a[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

a(n, k) = if(k == 1, 0, sumdiv(n, d, moebius(n/d) * numdiv(k^d-1))); \\ Amiram Eldar, Jan 25 2025

A(n,k) = |{m : multiplicative order of k mod m = n}|.

A(n,k) = Sum_{d|n} mu(n/d)*tau(k^d-1), mu = A008683, tau = A000005.

A059499 a(n) = |{m : multiplicative order of 2 mod m = n}|.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 3, 16, 1, 5, 5, 8, 1, 24, 1, 38, 9, 11, 3, 68, 6, 5, 4, 54, 7, 79, 1, 16, 11, 5, 13, 462, 3, 5, 13, 140, 3, 123, 7, 110, 54, 11, 7, 664, 2, 114, 29, 118, 7, 124, 59, 188, 13, 55, 3, 4456, 1, 5, 82, 96, 5, 353, 3, 118, 11, 485, 7

Offset: 1

Vladeta Jovovic, Feb 04 2001

nonn

Also, number of primitive factors of 2^n - 1 (cf. A212953). - Max Alekseyev, May 03 2022
The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m). See A002326.
a(n) is odd iff n is squarefree, A005117. - Thomas Ordowski, Jan 18 2014
The set S for which a(n) = |S| contains an odd number of prime powers p^k, where k > 0 and p == 3 (mod 4), iff n is squarefree and greater than one. - Isaac Saffold, Dec 28 2019

			a(3) = |{7}| = 1, a(4) = |{5,15}| = 2, a(6) = |{9,21,63}| = 3.

Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 200 terms from Alois P. Heinz)

Column k=2 of A212957.
Primitive factors of b^n - 1: this sequence (b=2), A059885 (b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A001037, A046801, A058943, A059912, A112927, A212953.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(2^d-1), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, May 31 2012

a[n_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, 2^d - 1], {d, Divisors[n]}]; Table[a[n], {n, 1, 71} ] (* Jean-François Alcover, Dec 12 2012 *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(2^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{d|n} A008683(n/d) * A046801(d) = Sum_{d|A007947(n)} A008683(d) * A046801(n/d). - Max Alekseyev, May 03 2022

a(n) = 1 iff 2^n-1 is noncomposite. a(prime(n)) = 2^A088863(n)-1. - Thomas Ordowski, Jan 16 2014

More terms from John W. Layman, Mar 22 2002

More terms from Alois P. Heinz, May 31 2012

A059885 a(n) = |{m : multiplicative order of 3 mod m = n}|.

Original entry on oeis.org

2, 2, 2, 6, 4, 10, 2, 14, 4, 16, 6, 58, 2, 10, 16, 88, 6, 108, 6, 150, 10, 54, 6, 290, 18, 10, 56, 138, 14, 716, 14, 144, 22, 118, 40, 1088, 6, 54, 90, 670, 14, 730, 6, 570, 356, 22, 30, 13864, 124, 342, 54, 138, 14, 3912, 116, 1362, 118, 238, 6, 22058, 6, 110

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

The multiplicative order of a mod m, GCD(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m). a(n) = number of orders of degree-n monic irreducible polynomials over GF(3).

Also, number of primitive factors of 3^n - 1 (cf. A218356). - Max Alekseyev, May 03 2022

			a(2) = |{4,8}| = 2, a(4) = |{5,10,16,20,40,80}| = 6, a(6) = |{7,14,28,52,56,91,104,182,364,728}| = 10.

Max Alekseyev, Table of n, a(n) for n = 1..690 (first 100 terms from Alois P. Heinz)

Primitive factors of b^n - 1: A059499 (b=2), this sequence (b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A027376, A057958, A058944, A074477, A143663, A212906, A218356.
Column k=3 of A212957.

with(numtheory); A059885 := proc(n) local d,s; s := 0; for d in divisors(n) do s := s+mobius(n/d)*tau(3^d-1); od; RETURN(s); end;

a[n_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, 3^d - 1], {d, Divisors[n]}]; Table[a[n], {n, 1, 62} ] (* Jean-François Alcover, Dec 12 2012 *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(3^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{ d divides n } mu(n/d)*tau(3^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A059887 a(n) = |{m : multiplicative order of 5 mod m=n}|.

Original entry on oeis.org

3, 5, 3, 12, 9, 37, 3, 28, 18, 47, 3, 180, 3, 53, 81, 176, 9, 446, 21, 564, 39, 117, 9, 884, 180, 53, 360, 244, 21, 5959, 9, 800, 39, 111, 369, 9536, 21, 483, 39, 5476, 9, 18289, 9, 1140, 2958, 111, 3, 9424, 6, 3852, 177, 884, 21, 81048, 561, 1188, 69, 227, 9

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m). a(n) = number of orders of degree-n monic irreducible polynomials over GF(5).

Also, number of primitive factors of 5^n - 1 (cf. A218357). - Max Alekseyev, May 03 2022

Max Alekseyev, Table of n, a(n) for n = 1..502

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), this sequence (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A001692, A053447, A057956, A058945, A074479, A143665, A212485, A218357
Column k=5 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(5^d-1), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 12 2012

a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 5^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 13 2024, after Alois P. Heinz *)

a(n) = sumdiv(n, d, moebius(n/d)*numdiv(5^d-1)); \\ Michel Marcus, Dec 13 2024

a(n) = Sum_{d|n} mu(n/d)*tau(5^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A059889 a(n) = |{m : multiplicative order of 7 mod m=n}|.

Original entry on oeis.org

4, 6, 8, 26, 4, 42, 12, 48, 52, 66, 12, 778, 4, 138, 80, 300, 12, 528, 12, 1430, 72, 138, 28, 15216, 24, 66, 1216, 966, 28, 3630, 28, 1344, 360, 58, 108, 16988, 28, 138, 176, 12752, 28, 7398, 12, 4422, 1900, 122, 12, 131028, 240, 536, 744, 1046, 28, 23744, 44

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).
a(n) = number of orders of degree n monic irreducible polynomials over GF(7).
Also, number of primitive factors of 7^n - 1 (cf. A218358). - Max Alekseyev, May 03 2022

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

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), this sequence (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A058946, A053450, A057954, A058946, A074249, A212486, A218358
Column k=7 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(7^d-1), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 12 2012

a[n_] := DivisorSum[n, MoebiusMu[n/#] * DivisorSigma[0, 7^#-1] &]; Array[a, 60] (* Amiram Eldar, Jan 25 2025 *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(7^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{d|n} mu(n/d)*tau(7^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A059886 a(n) = |{m : multiplicative order of 4 mod m=n}|.

Original entry on oeis.org

2, 2, 4, 4, 6, 16, 6, 8, 26, 38, 14, 68, 6, 54, 84, 16, 6, 462, 6, 140, 132, 110, 14, 664, 120, 118, 128, 188, 62, 4456, 6, 96, 364, 118, 498, 7608, 30, 118, 180, 568, 30, 9000, 30, 892, 3974, 494, 62, 5360, 24, 8024, 1524, 892, 62, 9600, 3050, 1784, 372, 446

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

The multiplicative order of a mod m, GCD(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).
a(n) is the number of orders of degree-n monic irreducible polynomials over GF(4).
Also, number of primitive factors of 4^n - 1. - Max Alekseyev, May 03 2022

			a(1) = |{1,3}| = 2, a(2) = |{5,15}| =2, a(3) = |{7,9,21,63}| =4, a(4) = |{17,51,85,255}| = 4.

Max Alekseyev, Table of n, a(n) for n = 1..1128 (first 160 terms from Alois P. Heinz)

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), this sequence (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A027377, A053447, A057957, A058948, A112092, A274906.
Column k=4 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(4^d-1), d=divisors(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 12 2012

a[n_] := DivisorSum[n, MoebiusMu[n/#]*DivisorSigma[0, 4^# - 1]&]; Array[a, 100] (* Jean-François Alcover, Nov 11 2015 *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(4^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{ d divides n } mu(n/d)*tau(4^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A059888 a(n) = |{m : multiplicative order of 6 mod m=n}|.

Original entry on oeis.org

2, 2, 2, 4, 4, 10, 2, 8, 12, 40, 6, 108, 6, 42, 40, 48, 30, 100, 6, 332, 10, 22, 30, 376, 26, 118, 48, 332, 2, 1436, 6, 448, 54, 222, 88, 7952, 62, 54, 54, 2680, 6, 698, 30, 476, 1476, 222, 14, 7632, 28, 438, 478, 1916, 14, 1872, 84, 11896, 118, 58, 14, 784452

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

The multiplicative order of a mod m, GCD(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).

Also, number of primitive factors of 6^n - 1. - Max Alekseyev, May 03 2022

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

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), this sequence (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Column k=6 of A212957.
Cf. A000005, A008683, A053449, A057955, A274907.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(6^d-1), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 12 2012

a[n_] := DivisorSum[n, MoebiusMu[n/#] * DivisorSigma[0, 6^#-1] &]; Array[a, 60] (* Amiram Eldar, Jan 25 2025 *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(6^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{ d divides n } mu(n/d)*tau(6^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A366654 a(n) = phi(8^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

6, 36, 432, 1728, 27000, 139968, 1778112, 6635520, 113467392, 534600000, 6963536448, 26121388032, 465193834560, 2427720325632, 28548223200000, 109586090557440, 1910296842179040, 9618417501143040, 123523151337020736, 406467072000000000, 7713001620195508224

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..500

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), this sequence (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A010705, A024088, A057953, A059890, A274908, A366651, A366652, A366653.

Mathematica
```
EulerPhi[8^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(8^n-1)}
```

from sympy import totient
def A366654(n): return totient((1<<3*n)-1) # Chai Wah Wu, Oct 15 2023

a(n) = A053287(3*n). - Max Alekseyev, Jan 09 2024

A366653 Sum of the divisors of 8^n-1.

Original entry on oeis.org

8, 104, 592, 8736, 38912, 473600, 2466048, 38054016, 155493536, 2015330304, 10359014400, 166290432000, 636328345600, 7645340651520, 42424026529792, 648494317126656, 2599936977797120, 32817383473149440, 164708609085669376, 3010983668199456768

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

			a(5)=38912 because 8^5-1 has divisors {1, 7, 31, 151, 217, 1057, 4681, 32767}.

Max Alekseyev, Table of n, a(n) for n = 1..500

Cf. A024088, A000203, A057953, A059890, A366651, A366652, A366654.

Cf. A075708, A366576, A366603, A366613, A366622, A366634, A366662, A102146, A366684, A366710.

a:=n->numtheory[sigma](8^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[1, 8^Range[30]-1]
```

[sigma(8**n-1, 1) for n in range(1, 21)] # Stefano Spezia, Aug 02 2025

a(n) = sigma(8^n-1) = A000203(A024088(n)).

a(n) = A075708(3*n). - Max Alekseyev, Jan 09 2024

cp's OEIS Frontend

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

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A212957 A(n,k) is the number of moduli m such that the multiplicative order of k mod m equals n; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A059499 a(n) = |{m : multiplicative order of 2 mod m = n}|.

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A059885 a(n) = |{m : multiplicative order of 3 mod m = n}|.

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A059887 a(n) = |{m : multiplicative order of 5 mod m=n}|.

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A059889 a(n) = |{m : multiplicative order of 7 mod m=n}|.

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A059886 a(n) = |{m : multiplicative order of 4 mod m=n}|.

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