A053450 -id:A053450 - cp's OEIS frontend

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

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).

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).

A050979 Haupt-exponents of 7 modulo integers relatively prime to 7.

Original entry on oeis.org

1, 1, 2, 4, 1, 2, 3, 4, 10, 2, 12, 4, 2, 16, 3, 3, 4, 10, 22, 2, 4, 12, 9, 7, 4, 15, 4, 10, 16, 6, 9, 3, 12, 4, 40, 6, 10, 12, 22, 23, 2, 4, 16, 12, 26, 9, 20, 3, 7, 29, 4, 60, 15, 8, 12, 10, 66, 16, 22, 70, 6, 24, 9, 4, 6, 12, 78, 4, 27, 40, 41, 16, 6, 7, 10, 88, 12, 22, 15, 23, 12

Offset: 1

Eric W. Weisstein

nonn

R. J. Mathar, Table of n, a(n) for n = 1..8571
Eric Weisstein's World of Mathematics, Multiplicative Order

Cf. A002326 (base 2), A002329, A050977 (base 5), A053450.

A054711 Multiplicative order of 11 mod n, where gcd(n, 11) = 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 2, 6, 1, 2, 12, 3, 2, 4, 16, 6, 3, 2, 6, 22, 2, 5, 12, 18, 6, 28, 2, 30, 8, 16, 3, 6, 6, 3, 12, 2, 40, 6, 7, 6, 22, 46, 4, 21, 5, 16, 12, 26, 18, 6, 6, 28, 58, 2, 4, 30, 6, 16, 12, 66, 16, 22, 3, 70, 6, 72, 6, 10, 6, 12, 39, 4, 54, 40, 41

Offset: 1

Henry Bottomley, Apr 20 2000

The original version "Number of powers of 11 modulo n" that was similar to A054703-A054717 is now in A351524. - Georg Fischer, Feb 13 2022

Cf. A053446 (of 3 mod n), A053448 (5), A053449 (6), A053450 (7), A053452 (9).

Cf. A351524.

MultiplicativeOrder[11, #] & /@ Select[ Range@ 90, GCD[11, #] == 1 &] (* Robert G. Wilson v, Apr 05 2011 *)

lista(nn) = {for(n=1, nn, if (gcd(n, 11) == 1, print1(znorder(Mod(11, n)), ", ")););} \\ Michel Marcus, Feb 09 2015

Corrected by Michel Marcus, Feb 11 2015

cp's OEIS Frontend

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

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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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A050979 Haupt-exponents of 7 modulo integers relatively prime to 7.

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A054711 Multiplicative order of 11 mod n, where gcd(n, 11) = 1.

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