A218356 -id:A218356 - cp's OEIS frontend

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

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).

A143663 a(n) is the least prime such that the multiplicative order of 3 mod a(n) equals n, or a(n)=1 if no such prime exists.

Original entry on oeis.org

2, 1, 13, 5, 11, 7, 1093, 41, 757, 61, 23, 73, 797161, 547, 4561, 17, 1871, 19, 1597, 1181, 368089, 67, 47, 6481, 8951, 398581, 109, 29, 59, 31, 683, 21523361, 2413941289, 103, 71, 530713, 13097927, 2851, 313, 42521761, 83, 43, 431, 5501, 181, 23535794707

Offset: 1

Vladimir Shevelev, Aug 28 2008

nonn

If a(n) differs from 1, then a(n) is the minimal prime divisor of A064079(n).

Max Alekseyev, Table of n, a(n) for n = 1..730 (first 153 terms from Robert G. Wilson v)

Cf. A002326, A064079, A112092, A218356, A057958, A074477.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

a:= proc(n) local f,p;
f:= numtheory:-factorset(3^n - 1);
for  p in f do
   if numtheory:-order(3,p) = n then return p fi
od:
1
end proc:
seq(a(n),n=1..100); # Robert Israel, Oct 13 2014

p = 2; t = Table[0, {100}]; While[p < 100000001, a = MultiplicativeOrder[3, p]; If[0 < a < 101 && t[[a]] == 0, t[[a]] = p; Print[{a, p}]];  p = NextPrime@ p]; t (* Robert G. Wilson v, Oct 13 2014 *)

More terms from Robert G. Wilson v, Dec 11 2013

A003060 Smallest number with reciprocal of period length n in decimal (base 10).

Original entry on oeis.org

1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, 21649, 707, 53, 2629, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 511, 21401, 583, 243, 29, 3191, 211, 2791, 353, 67, 103, 71, 1919, 2028119, 909090909090909091

Offset: 0

N. J. A. Sloane

For n > 0, a(n) is the least divisor d > 1 of 10^n - 1 such that the multiplicative order of 10 mod d is n. For prime n > 3, a(n) = A007138(n). - T. D. Noe, Aug 07 2007

For n > 1, a(n) is the smallest positive d such that d divides 10^n - 1 and does not divide any of 10^k - 1 for 0 < k < n. - Maciej Ireneusz Wilczynski, Sep 06 2012, corrected by M. F. Hasler, Jun 28 2022. (For n = 1, d = 1 divides 10^n - 1 and does not divide any 10^k - 1 with 0 < k < n, but a(1) = 3 > 1.)

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
"Cycle lengths of reciprocals", Popular Computing (Calabasas, CA), Vol. 1 (No. 4, Jul 1973), pp. 12-14.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Smallest primitive divisors of b^n-1: A212953 (b=2), A218356 (b=3), A218357 (b=5), A218358 (b=7), this sequence (b=10), A218359 (b=11), A218360 (b=13), A218361 (b=17), A218362 (b=19), A218363 (b=23), A218364 (b=29).

Cf. A007138, A057951, A005422, A176973.

a[n_] := First[ Select[ Divisors[10^n - 1], MultiplicativeOrder[10, #] == n &, 1]]; a[0] = 1; a[1] = 3; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jul 13 2012, after T. D. Noe *)

apply( {A003060(n)=!fordiv(10^n-!!n, d, d>1 && znorder(Mod(10,d))==n && return(d))}, [0..50]) \\ M. F. Hasler, Jun 28 2022

Comment corrected by T. D. Noe, Apr 15 2010
More terms from T. D. Noe, Apr 15 2010
b-file truncated at uncertain term a(439) by Max Alekseyev, Apr 30 2022

A085028 Number of prime factors of cyclotomic(n,3), which is A019321(n), the value of the n-th cyclotomic polynomial evaluated at x=3.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 1, 2, 2, 1, 2, 1, 3, 2, 3, 2, 3, 2, 1, 3, 2, 1, 2, 2, 4, 1, 3, 3, 2, 2, 3, 1, 4, 3, 5, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 1, 2, 2, 1, 2, 3, 2, 3, 2, 2, 1, 1, 1, 4, 3, 3, 2, 3, 4, 3, 2, 3, 2, 4, 2, 2, 1, 3, 3, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 4

Offset: 1

T. D. Noe, Jun 19 2003

nonn

The Mobius transform of this sequence yields A057958, number of prime factors of 3^n-1.

See references at A085021

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

omega(Phi(n,x)): A085021 (x=2), this sequence (x=3), A085029 (x=4), A085030 (x=5), A085031 (x=6), A085032 (x=7), A085033 (x=8), A085034 (x=9), A085035 (x=10).

Cf. A019321, A057958, A059885, A143663, A218356, A074477.

Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 3]]][[2]], {n, 1, 100}]

A213224 Minimal order A(n,k) of degree-n irreducible polynomials over GF(prime(k)); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 7, 1, 3, 13, 5, 1, 4, 31, 5, 31, 1, 3, 9, 13, 11, 9, 1, 7, 7, 5, 11, 7, 127, 1, 3, 9, 16, 2801, 7, 1093, 17, 1, 4, 307, 5, 25, 36, 19531, 32, 73, 1, 3, 27, 5, 30941, 9, 29, 32, 757, 11, 1, 3, 7, 16, 88741, 63, 43, 64, 19, 44, 23

Offset: 1

Alois P. Heinz, Jun 06 2012

Maximal order of degree-n irreducible polynomials over GF(prime(k)) is prime(k)^n-1 and thus A(n,k) < prime(k)^n.

			A(4,1) = 5: The degree-4 irreducible polynomials p over GF(prime(1)) = GF(2) are 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. Their orders (i.e., the smallest integer e for which p divides x^e+1) are 5, 15, 15, and the minimal order is 5. (1+x+x^2+x^3+x^4) * (1+x) == x^5+1 (mod 2).
Square array A(n,k) begins:
    1,    1,     1,    1,   1,       1,        1,   1, ...
    3,    4,     3,    4,   3,       7,        3,   4, ...
    7,   13,    31,    9,   7,       9,      307,  27, ...
    5,    5,    13,    5,  16,       5,        5,  16, ...
   31,   11,    11, 2801,  25,   30941,    88741, 151, ...
    9,    7,     7,   36,   9,      63,        7,   7, ...
  127, 1093, 19531,   29,  43, 5229043, 25646167, 701, ...
   17,   32,    32,   64,  32,      32,      128,  17, ...

Alois P. Heinz, Antidiagonals n = 1..45, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Columns k=1-10 give: A212953, A218356, A218357, A218358, A218359, A218360, A218361, A218362, A218363, A218364.
Columns k=1-10 are first columns of: A059912, A212906, A212485, A212486, A218336, A218337, A218338, A218339, A218340, A218341.
Cf. A212737 (all orders).

with(numtheory):
M:= proc(n, i) option remember;
      divisors(ithprime(i)^n-1) minus U(n-1, i)
    end:
U:= proc(n, i) option remember;
      `if`(n=0, {}, M(n, i) union U(n-1, i))
    end:
A:= (n, k)-> min(M(n, k)[]):
seq(seq(A(n, d+1-n), n=1..d), d=1..14);

M[n_, i_] := M[n, i] = Divisors[Prime[i]^n - 1] ~Complement~ U[n-1, i]; U[n_, i_] := U[n, i] = If[n == 0, {}, M[n, i] ~Union~ U[n-1, i]]; A[n_, k_] := Min[M[n, k]]; Table[Table[A[n, d+1-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A(n,k) = min(M(n,k)) with M(n,k) = {d : d|(prime(k)^n-1)} \ U(n-1,k) and U(n,k) = M(n,k) union U(n-1,k) for n>0, U(0,k) = {}.

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).

cp's OEIS Frontend

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

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A143663 a(n) is the least prime such that the multiplicative order of 3 mod a(n) equals n, or a(n)=1 if no such prime exists.

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A003060 Smallest number with reciprocal of period length n in decimal (base 10).

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A085028 Number of prime factors of cyclotomic(n,3), which is A019321(n), the value of the n-th cyclotomic polynomial evaluated at x=3.

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A213224 Minimal order A(n,k) of degree-n irreducible polynomials over GF(prime(k)); square array A(n,k), n>=1, k>=1, read by antidiagonals.

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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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