A059499 -id:A059499 - cp's OEIS frontend

A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 2, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 4

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A046051, the number of prime factors of Mersenne number 2^n-1.

The number of prime factors in the primitive part of 2^n-1. - T. D. Noe, Jul 19 2008

			a(11) = 2 because cyclotomic(11,2) = 2047, which has two factors: 23 and 89.

Max Alekseyev, Table of n, a(n) for n = 1..1206 (first 500 term from T. D. Noe)
Vjekoslav Kovač and Florian Luca, On the number of divisors of Mersenne numbers, arXiv:2506.04883 [math.NT], 2025. See Table 2 p. 9.
H. Mishima, Factorization of Cyclotomic Numbers
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Eric Weisstein's World of Mathematics, Möbius Transform

omega(Phi(n,x)): this sequence (x=2), A085028 (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A005420, A019320, A046051, A046801, A059499, A064078, A112927, A212953.

Join[{0}, Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 2]]][[2]], {n, 2, 100}]]

a(n) = #factor(polcyclo(n, 2))~; \\ Michel Marcus, Mar 06 2015

A059891 a(n) = |{m : multiplicative order of 9 mod m = n}|.

Original entry on oeis.org

4, 6, 12, 14, 20, 58, 12, 88, 112, 150, 60, 290, 12, 138, 732, 144, 124, 1088, 60, 670, 740, 570, 28, 13864, 360, 138, 3968, 1362, 252, 22058, 124, 320, 1972, 1146, 732, 10704, 124, 570, 12260, 15176, 124, 60470, 28, 11634, 195728, 282, 508, 116592, 2032

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(9).
Also, number of primitive factors of 9^n - 1. - Max Alekseyev, May 03 2022

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

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), this sequence (b=9), A059892 (b=10).
Cf. A000005, A008683, A027381, A053452, A057952, A058946, A274909.
Column k=9 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(9^d-1), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 12 2012

a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 9^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(9^d-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = Sum_{d|n} mu(n/d)*tau(9^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A086251 Number of primitive prime factors of 2^n - 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3

Offset: 1

T. D. Noe, Jul 14 2003

A prime factor of 2^n - 1 is called primitive if it does not divide 2^r - 1 for any r < n. Equivalently, p is a primitive prime factor of 2^n - 1 if ord(2,p) = n. Zsigmondy's theorem says that there is at least one primitive prime factor for n > 1, except for n=6. See A086252 for those n that have a record number of primitive prime factors.
Number of odd primes p such that A002326((p-1)/2) = n. Number of occurrences of number n in A014664. - Thomas Ordowski, Sep 12 2017
The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - Jeppe Stig Nielsen, Sep 01 2020

			a(11) = 2 because 2^11 - 1 = 23*89 and both 23 and 89 have order 11.

Table of n, a(n) for n = 1..1206
Eric Weisstein's World of Mathematics, Zsigmondy Theorem

Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252, A002588, A005420, A002184, A046801, A049093, A049094, A059499, A085021, A097406, A112927, A237043.

Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]

a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ Michel Marcus, Sep 12 2017

a(n) = my(m=polcyclo(n, 2)); omega(m/gcd(m,n)) \\ Jeppe Stig Nielsen, Sep 01 2020

a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
a(n) <= A182590(n). - Thomas Ordowski, Sep 14 2017
a(n) = A001221(A064078(n)). - Thomas Ordowski, Oct 26 2017

Terms to a(500) in b-file from T. D. Noe, Nov 11 2010
Terms a(501)-a(1200) in b-file from Charles R Greathouse IV, Sep 14 2017
Terms a(1201)-a(1206) in b-file from Max Alekseyev, Sep 11 2022

A212906 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(3) listed in ascending order.

Original entry on oeis.org

1, 2, 4, 8, 13, 26, 5, 10, 16, 20, 40, 80, 11, 22, 121, 242, 7, 14, 28, 52, 56, 91, 104, 182, 364, 728, 1093, 2186, 32, 41, 82, 160, 164, 205, 328, 410, 656, 820, 1312, 1640, 3280, 6560, 757, 1514, 9841, 19682, 44, 61, 88, 122, 244, 484, 488, 671, 968, 1342

Offset: 1

Boris Putievskiy, May 29 2012

The elements m of row n, are also solutions to the equation: multiplicative order of 3 mod m = n, with gcd(m,3) = 1, cf. A053446.

			Triangle T(n,k) begins:
1,   2;
4,   8;
13, 26;
5,  10,  16,  20, 40, 80;
11, 22, 121, 242;
7,  14,  28,  52, 56, 91, 104, 182, 364, 728;

R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 555-557.
V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

Alois P. Heinz, Rows n = 1..47, flattened (first 13 rows from Boris Putievskiy)
Eric Weisstein's World of Mathematics, Irreducible Polynomial
XIAO, Polynomial order (computes the order of an irreducible polynomial over a finite field GF(p))

Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.
Column k=2 of A212737.
Column k=1 gives: A218356.

with(numtheory):
M:= proc(n) option remember;
      divisors(3^n-1) minus U(n-1)
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..15);  # Alois P. Heinz, Jun 02 2012

M[n_] := M[n] = Divisors[3^n - 1] ~Complement~ U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]]; Array[T, 15] // Flatten (* Jean-François Alcover, Jun 10 2018, after Alois P. Heinz *)

T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (3^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(3^d-1) = A059885(n).

A059907 a(n) = |{m : multiplicative order of n mod m = 2}|.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 6, 4, 6, 3, 12, 2, 10, 6, 8, 4, 13, 2, 18, 6, 10, 4, 16, 4, 12, 9, 12, 4, 26, 2, 20, 6, 8, 12, 20, 4, 15, 6, 16, 4, 32, 2, 24, 10, 10, 6, 20, 4, 26, 9, 18, 4, 26, 6, 32, 12, 12, 4, 28, 2, 20, 10, 12, 18, 25, 4, 24, 6, 26, 4, 52, 2, 18, 10, 12, 18, 26, 4, 40, 8, 14, 5, 28

Offset: 1

Vladeta Jovovic, Feb 08 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).

			a(2) = |{3}| = 1, a(3) = |{4,8}| = 2, a(4) = |{5,15}| = 2, a(5) = |{3,6,8,12,24}| = 5, a(6) = |{7,35}| = 2, a(7) = |{4,8,12,16,24,48}| = 6,...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002329, A059908-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A055205, A048691, A048785, A347191.

Row n=2 of A212957. - Alois P. Heinz, Oct 24 2012

with(numtheory):f := n->tau(n^2-1)-tau(n-1):for n from 1 to 100 do printf(`%d,`,f(n)) od:

a[n_] := Subtract @@ DivisorSigma[0, {n^2-1, n-1}]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)

a(n) = if(n == 1, 0, numdiv(n^2-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^2-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A212485 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(5) listed in ascending order.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 12, 24, 31, 62, 124, 13, 16, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624, 11, 22, 44, 71, 142, 284, 781, 1562, 3124, 7, 9, 14, 18, 21, 28, 36, 42, 56, 63, 72, 84, 93, 126, 168, 186, 217, 248, 252, 279, 372, 434, 504, 558, 651, 744, 868, 1116

Offset: 1

Boris Putievskiy, Jun 02 2012

The elements m of row n, are also solutions to the equation: multiplicative order of 5 mod m = n, with gcd(m,5) = 1, cf. A050977.

			Triangle T(n,k) begins:
   1,  2,   4;
   3,  6,   8, 12,  24;
  31, 62, 124;
  13, 16,  26, 39,  48,  52,  78,  104,  156, 208, 312, 624;
  11, 22,  44, 71, 142, 284, 781, 1562, 3124;
  ...

R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 557-560.

Alois P. Heinz, Rows n = 1..32, flattened
V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

Cf. A212906, A059912, A058944, A059499, A059886-A059892.
Column k=3 of A212737.
Column k=1 gives: A218357.

with(numtheory):
M:= proc(n) option remember;
      `if`(n=1, {1, 2, 4}, divisors(5^n-1) minus U(n-1))
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..8);

M[n_] := M[n] = If[n == 1, {1, 2, 4}, Divisors[5^n-1] ~Complement~ U[n-1]];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]]; Array[T, 8] // Flatten (* Jean-François Alcover, Jun 10 2018, from Maple *)

T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (5^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,4}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(5^d-1) = A059887.

A212486 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 16, 24, 48, 9, 18, 19, 38, 57, 114, 171, 342, 5, 10, 15, 20, 25, 30, 32, 40, 50, 60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800, 1200, 2400, 2801, 5602, 8403, 16806, 36, 43, 72, 76, 86, 129, 144, 152, 172, 228, 258

Offset: 1

Boris Putievskiy, Jun 02 2012

The elements m of row n, are also solutions to the equation: multiplicative order of 7 mod m = n, with gcd(m,7) = 1, cf. A053450.

			Triangle T(n,k) begins:
  1,  2,  3,  6;
  4,  8, 12, 16, 24,  48;
  9, 18, 19, 38, 57, 114, 171, 342;
  5, 10, 15, 20, 25,  30,  32,  40, 50, 60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800, 1200, 2400;
  ...

R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 560-562.
V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

Boris Putievskiy, Table of n, a(n) for n = 1..102
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Gang Xiao, (computes the order of an irreducible polynomial over a finite field GF(p))

Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.
Column k=4 of A212737.
Column k=1 gives: A218358.

with(numtheory):
M:= proc(n) option remember;
      `if`(n=1, {1, 2, 3, 6}, divisors(7^n-1) minus U(n-1))
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..7);

M[n_] := M[n] = If[n == 1, {1, 2, 3, 6}, Divisors[7^n - 1] ~Complement~ U[n - 1]];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]];
Table[T[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Sep 24 2022, from Maple code *)

T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (7^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,3,6}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(7^d-1) = A059889(n).

A059911 a(n) = |{m : multiplicative order of n mod m = 6}|.

Original entry on oeis.org

0, 3, 10, 16, 37, 10, 42, 24, 58, 53, 164, 26, 68, 38, 32, 68, 169, 22, 222, 38, 42, 50, 328, 40, 180, 219, 108, 26, 334, 82, 460, 82, 92, 72, 220, 108, 449, 86, 128, 80, 192, 22, 336, 110, 222, 218, 540, 84, 778, 129, 150, 80, 270, 54, 328, 356, 132, 68, 348, 22

Offset: 1

Vladeta Jovovic, Feb 08 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).

			a(2) = |{9,21,63}| = 3, a(3) = |{7,14,28,52,56,91,104,182,364,728}| = 10, a(4) = |{13,35,39,45,65,91,105,117,195,273,315,455,585,819,1365,4095}| = 16,...

Cf. A059907-A059910, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

a[n_] := Total[{1, -1, -1, 1} * DivisorSigma[0, n^{6, 3, 2, 1} - 1]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025*)

a(n) = if(n == 1, 0, numdiv(n^6-1) - numdiv(n^3-1) - numdiv(n^2-1) + numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^6-1)-tau(n^3-1)-tau(n^2-1)+tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A059908 a(n) = |{m : multiplicative order of n mod m = 3}|.

Original entry on oeis.org

0, 1, 2, 4, 3, 2, 8, 2, 12, 5, 12, 2, 12, 2, 4, 20, 5, 6, 10, 2, 6, 14, 12, 2, 40, 9, 4, 6, 18, 10, 16, 6, 6, 8, 12, 12, 39, 2, 12, 8, 8, 6, 16, 6, 18, 26, 12, 6, 50, 3, 18, 8, 18, 2, 32, 12, 8, 20, 4, 6, 60, 2, 12, 26, 21, 4, 64, 10, 6, 8, 8, 6, 20, 14, 4, 12, 6, 4, 64, 2, 70, 7, 12, 6, 24

Offset: 1

Vladeta Jovovic, Feb 08 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).

			a(2) = |{7}| = 1, a(3) = |{13,26}| = 2, a(4) = |{7,9,21,63}| = 4, a(5) = |{31,62,124}| = 3, a(6) = |{43,215}| = 2, a(7) = |{9,18,19,38,57,114,171,342}| = 8,...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A059907, A059909-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

Row n=3 of A212957. - Alois P. Heinz, Oct 24 2012

Table[DivisorSigma[0,n^3-1]-DivisorSigma[0,n-1],{n,90}] (* Harvey P. Dale, Feb 03 2015 *)

a(n) = if(n == 1, 0, numdiv(n^3-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^3-1)-tau(n-1), where tau(n) = number of divisors of n A000005. Generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

A059909 a(n) = |{m : multiplicative order of n mod m = 4}|.

Original entry on oeis.org

0, 2, 6, 4, 12, 4, 26, 18, 14, 6, 24, 12, 64, 8, 16, 8, 66, 20, 36, 8, 64, 24, 76, 6, 28, 12, 64, 24, 48, 12, 100, 40, 50, 48, 36, 8, 96, 40, 28, 8, 104, 12, 208, 36, 24, 36, 200, 18, 48, 36, 36, 24, 128, 8, 152, 16, 172, 24, 48, 12, 48, 36, 56, 72, 40, 8, 128, 56, 48, 40

Offset: 1

Vladeta Jovovic, Feb 08 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).

			a(2) = |{5, 15}| = 2, a(3) = |{5, 10, 16, 20, 40, 80}| = 6, a(4) = |{17, 51, 85, 255}| = 4, a(5) = |{13, 16, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624}| = 12, ...

Cf. A059907, A059908, A059910-A059911, A059499, A059885-A059892, A002326, A053446-A053452, A002329, A055205, A048691, A048785.

Table[DivisorSigma[0,n^4-1]-DivisorSigma[0,n^2-1],{n,70}] (* Harvey P. Dale, Nov 30 2011 *)

a(n) = if(n == 1, 0, numdiv(n^4-1) - numdiv(n^2-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^4-1)-tau(n^2-1), where tau(n) = number of divisors of n A000005. More generally, if b(n, r) = |{m : multiplicative order of n mod m = r}| then b(n, r) = Sum_{d|r} mu(d)*tau(n^(r/d)-1), where mu(n) = Moebius function A008683.

cp's OEIS Frontend

A085021 Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.

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A059891 a(n) = |{m : multiplicative order of 9 mod m = n}|.

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A086251 Number of primitive prime factors of 2^n - 1.

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A212906 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(3) listed in ascending order.

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A059907 a(n) = |{m : multiplicative order of n mod m = 2}|.

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A212485 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(5) listed in ascending order.

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A212486 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order.

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