A050977 -id:A050977 - cp's OEIS frontend

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

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.

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.

A053448 Multiplicative order of 5 mod m, where gcd(m, 5) = 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 2, 6, 5, 2, 4, 6, 4, 16, 6, 9, 6, 5, 22, 2, 4, 18, 6, 14, 3, 8, 10, 16, 6, 36, 9, 4, 20, 6, 42, 5, 22, 46, 4, 42, 16, 4, 52, 18, 6, 18, 14, 29, 30, 3, 6, 16, 10, 22, 16, 22, 5, 6, 72, 36, 9, 30, 4, 39, 54, 20, 82, 6, 42, 14, 10, 44, 12, 22, 6, 46, 8, 96, 42, 30, 25, 16

Offset: 1

David W. Wilson

nonn

Essentially the same as A050977. - R. J. Mathar, Oct 21 2012

Cf. A047201, A002326 (order of 2), A053446 (order of 3), A053447 (order of 4).

MultiplicativeOrder[5, #] & /@ Select[ Range@ 100, GCD[5, #] == 1 &] (* Robert G. Wilson v, Apr 05 2011 *)

lista(nn) = {for(n=1, nn, if (gcd(n, 5) == 1, print1(znorder(Mod(5, n)), ", ")););} \\ Michel Marcus, Feb 09 2015

a(n) = multiplicative order of 5 modulo floor((5*n-1)/4), for n >= 1. This modulus is A047201(n). - Wolfdieter Lang, Sep 30 2020

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

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A053448 Multiplicative order of 5 mod m, where gcd(m, 5) = 1.

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