A212957 -id:A212957 - cp's OEIS frontend

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

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.

A218336 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(11) listed in ascending order.

Original entry on oeis.org

1, 2, 5, 10, 3, 4, 6, 8, 12, 15, 20, 24, 30, 40, 60, 120, 7, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330, 16, 48, 61, 80, 122, 183, 240, 244, 305, 366, 488, 610, 732, 915, 976, 1220, 1464, 1830, 2440, 2928, 3660, 4880, 7320, 14640, 25, 50, 3221, 6442

Offset: 1

Alois P. Heinz, Oct 26 2012

			Triangle begins:
   1,  2,    5,   10;
   3,  4,    6,    8,    12,    15,    20,     24,  30,  40, ...
   7, 14,   19,   35,    38,    70,    95,    133, 190, 266, ...
  16, 48,   61,   80,   122,   183,   240,    244, 305, 366, ...
  25, 50, 3221, 6442, 16105, 32210, 80525, 161050;
  ...

Alois P. Heinz, Rows n = 1..23, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Column k=5 of A212737.
Last elements of rows give: A024127.
Column k=1 gives: A218359.
Row lengths are A212957(n,11).

with(numtheory):
M:= proc(n) M(n):= divisors(11^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);

M[n_] := M[n] = Divisors[11^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, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

T(n,k) = k-th smallest element of M(n) = {d : d|(11^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218337 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(13) listed in ascending order.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 7, 8, 14, 21, 24, 28, 42, 56, 84, 168, 9, 18, 36, 61, 122, 183, 244, 366, 549, 732, 1098, 2196, 5, 10, 15, 16, 17, 20, 30, 34, 35, 40, 48, 51, 60, 68, 70, 80, 85, 102, 105, 112, 119, 120, 136, 140, 170, 204, 210, 238, 240, 255, 272, 280, 336

Offset: 1

Alois P. Heinz, Oct 26 2012

			Triangle begins:
:     1,     2,     3,      4,      6,     12;
:     7,     8,    14,     21,     24,     28,  42,  56,  84, 168;
:     9,    18,    36,     61,    122,    183, 244, 366, 549, ...
:     5,    10,    15,     16,     17,     20,  30,  34,  35, ...
: 30941, 61882, 92823, 123764, 185646, 371292;

Alois P. Heinz, Rows n = 1..20, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Column k=6 of A212737.
Column k=1 gives: A218360.
Row lengths are A212957(n,13).

with(numtheory):
M:= proc(n) M(n):= divisors(13^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);

M[n_] := Divisors[13^n-1] ~Complement~ U[n-1]; U[n_] := If[n == 0, {}, M[n] ~Union~  U[n-1]]; T[n_] := Sort[M[n]]; Table[T[n], {n, 1, 5}] // Flatten (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

T(n,k) = k-th smallest element of M(n) = {d : d|(13^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218338 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(17) listed in ascending order.

Original entry on oeis.org

1, 2, 4, 8, 16, 3, 6, 9, 12, 18, 24, 32, 36, 48, 72, 96, 144, 288, 307, 614, 1228, 2456, 4912, 5, 10, 15, 20, 29, 30, 40, 45, 58, 60, 64, 80, 87, 90, 116, 120, 145, 160, 174, 180, 192, 232, 240, 261, 290, 320, 348, 360, 435, 464, 480, 522, 576, 580, 696, 720

Offset: 1

Alois P. Heinz, Oct 26 2012

			Triangle begins:
      1,      2,      4,      8,      16;
      3,      6,      9,     12,      18,  24, 32, 36, 48, 72, ...
    307,    614,   1228,   2456,    4912;
      5,     10,     15,     20,      29,  30, 40, 45, 58, 60, ...
  88741, 177482, 354964, 709928, 1419856;

Alois P. Heinz, Rows n = 1..19, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Column k=7 of A212737.
Column k=1 gives: A218361.
Row lengths are A212957(n,17).

with(numtheory):
M:= proc(n) M(n):= divisors(17^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);

M[n_] := M[n] = Divisors[17^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, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

T(n,k) = k-th smallest element of M(n) = {d : d|(17^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218339 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(19) listed in ascending order.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 4, 5, 8, 10, 12, 15, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360, 27, 54, 127, 254, 381, 762, 1143, 2286, 3429, 6858, 16, 48, 80, 144, 181, 240, 362, 543, 720, 724, 905, 1086, 1448, 1629, 1810, 2172, 2715, 2896, 3258, 3620, 4344, 5430

Offset: 1

Alois P. Heinz, Oct 26 2012

			Triangle begins:
    1,   2,   3,   6,   9,   18;
    4,   5,   8,  10,  12,   15,   20,   24,   30,   36,   40, ...
   27,  54, 127, 254, 381,  762, 1143, 2286, 3429, 6858;
   16,  48,  80, 144, 181,  240,  362,  543,  720,  724,  905, ...
  151, 302, 453, 906, 911, 1359, 1822, 2718, 2733, 5466, 8199, ...
  ...

Alois P. Heinz, Rows n = 1..18, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Column k=8 of A212737.
Column k=1 gives: A218362.
Row lengths are A212957(n,19).

with(numtheory):
M:= proc(n) M(n):= divisors(19^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);

M[n_] := M[n] = Divisors[19^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, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

T(n,k) = k-th smallest element of M(n) = {d : d|(19^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218340 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(23) listed in ascending order.

Original entry on oeis.org

1, 2, 11, 22, 3, 4, 6, 8, 12, 16, 24, 33, 44, 48, 66, 88, 132, 176, 264, 528, 7, 14, 77, 79, 154, 158, 553, 869, 1106, 1738, 6083, 12166, 5, 10, 15, 20, 30, 32, 40, 53, 55, 60, 80, 96, 106, 110, 120, 159, 160, 165, 212, 220, 240, 265, 318, 330, 352, 424, 440

Offset: 1

Alois P. Heinz, Oct 26 2012

			Triangle begins:
       1,      2,      11,      22;
       3,      4,       6,       8,  12,  16,  24,  33,   44, ...
       7,     14,      77,      79, 154, 158, 553, 869, 1106, ...
       5,     10,      15,      20,  30,  32,  40,  53,   55, ...
  292561, 585122, 3218171, 6436342;
  ...

Alois P. Heinz, Rows n = 1..17, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Column k=9 of A212737.
Column k=1 gives: A218363.
Row lengths are A212957(n,23).

with(numtheory):
M:= proc(n) M(n):= divisors(23^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);

M[n_] := M[n] = Divisors[23^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, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

T(n,k) = k-th smallest element of M(n) = {d : d|(23^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218341 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(29) listed in ascending order.

Original entry on oeis.org

1, 2, 4, 7, 14, 28, 3, 5, 6, 8, 10, 12, 15, 20, 21, 24, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840, 13, 26, 52, 67, 91, 134, 182, 268, 364, 469, 871, 938, 1742, 1876, 3484, 6097, 12194, 24388, 16, 48, 80, 112, 240, 336, 421, 560

Offset: 1

Alois P. Heinz, Oct 26 2012

			Triangle begins:
       1,       2,       4,       7,       14,       28;
       3,       5,       6,       8,       10,       12,  15, ...
      13,      26,      52,      67,       91,      134, 182, ...
      16,      48,      80,     112,      240,      336, 421, ...
  732541, 1465082, 2930164, 5127787, 10255574, 20511148;
  ...

Alois P. Heinz, Rows n = 1..16, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Column k=10 of A212737.
Column k=1 gives: A218364.
Row lengths are A212957(n,29).

with(numtheory):
M:= proc(n) M(n):= divisors(29^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..5);

M[n_] := M[n] = Divisors[29^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, 5}] // Flatten (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

T(n,k) = k-th smallest element of M(n) = {d : d|(29^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}.

A218256 a(n) = |{m : multiplicative order of n mod m = 7}|.

Original entry on oeis.org

0, 1, 2, 6, 3, 2, 12, 10, 12, 9, 12, 6, 6, 2, 8, 60, 5, 6, 18, 14, 42, 8, 12, 14, 56, 3, 12, 12, 12, 14, 8, 14, 18, 12, 12, 44, 27, 2, 12, 4, 24, 6, 40, 14, 42, 6, 12, 6, 150, 5, 18, 60, 18, 14, 24, 4, 40, 12, 60, 2, 12, 6, 12, 138, 49, 4, 24, 2, 18, 12, 40, 2

Offset: 1

Alois P. Heinz, Oct 24 2012

nonn

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

Row n=7 of A212957.

Cf. A000005, A008683.

with(numtheory):
a:= n-> add(mobius(7/d) *tau(n^d-1), d={1, 7}):
seq(a(n), n=1..80);

a[n_] := Subtract @@ DivisorSigma[0, {n^7-1, n-1}]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)

a(n) = if(n == 1, 0, numdiv(n^7-1) - numdiv(n-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^7-1)-tau(n-1), with tau = A000005.

A218257 a(n) = |{m : multiplicative order of n mod m = 8}|.

Original entry on oeis.org

0, 4, 14, 8, 28, 8, 48, 72, 88, 36, 56, 48, 112, 48, 100, 16, 108, 72, 228, 16, 112, 96, 128, 12, 176, 72, 304, 32, 112, 48, 448, 144, 224, 64, 84, 48, 456, 144, 64, 48, 528, 48, 2064, 336, 152, 48, 800, 24, 300, 144, 228, 96, 608, 16, 704, 32, 256, 96, 688

Offset: 1

Alois P. Heinz, Oct 24 2012

nonn

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

Row n=8 of A212957.

Cf. A000005, A008683.

with(numtheory):
a:= n-> add(mobius(8/d) *tau(n^d-1), d={4, 8}):
seq(a(n), n=1..80);

a[n_] := Subtract @@ DivisorSigma[0, {n^8-1, n^4-1}]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jan 25 2025 *)

a(n) = if(n == 1, 0, numdiv(n^8-1) - numdiv(n^4-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^8-1)-tau(n^4-1), with tau = A000005.

A218258 a(n) = |{m : multiplicative order of n mod m = 9}|.

Original entry on oeis.org

0, 2, 4, 26, 18, 12, 52, 4, 112, 12, 16, 12, 30, 12, 24, 488, 30, 24, 64, 4, 12, 78, 48, 28, 464, 12, 56, 62, 72, 12, 104, 56, 36, 52, 48, 112, 432, 28, 48, 52, 48, 24, 488, 56, 72, 288, 48, 24, 580, 6, 24, 116, 360, 44, 344, 16, 48, 104, 24, 8, 312, 44, 112

Offset: 1

Alois P. Heinz, Oct 24 2012

nonn

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

Row n=9 of A212957.

Cf. A000005, A008683.

with(numtheory):
a:= n-> add(mobius(9/d) *tau(n^d-1), d={3, 9}):
seq(a(n), n=1..80);

Table[DivisorSigma[0,n^9-1]-DivisorSigma[0,n^3-1],{n,70}] (* Harvey P. Dale, Mar 15 2018 *)

a(n) = if(n == 1, 0, numdiv(n^9-1) - numdiv(n^3-1)); \\ Amiram Eldar, Jan 25 2025

a(n) = tau(n^9-1)-tau(n^3-1), with tau = A000005.

cp's OEIS Frontend

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

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A218336 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(11) listed in ascending order.

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A218337 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(13) listed in ascending order.

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A218338 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(17) listed in ascending order.

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A218339 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(19) listed in ascending order.

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Mathematica

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A218340 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(23) listed in ascending order.

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Maple

Mathematica

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A218341 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(29) listed in ascending order.

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Maple

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A218256 a(n) = |{m : multiplicative order of n mod m = 7}|.