A218358 -id:A218358 - 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).

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

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) = {}.

A366632 Number of distinct prime divisors of 7^n - 1.

Original entry on oeis.org

2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 4, 7, 3, 6, 6, 6, 4, 7, 4, 8, 6, 6, 5, 11, 5, 5, 9, 8, 5, 10, 5, 8, 8, 5, 7, 11, 5, 6, 7, 11, 5, 11, 4, 10, 10, 6, 4, 14, 8, 8, 9, 8, 5, 12, 6, 13, 8, 6, 6, 17, 6, 8, 9, 11, 9, 13, 6, 9, 9, 15, 4, 18, 7, 7, 10, 8, 9, 13, 4, 16, 13

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..388

Cf. A024075, A001221, A057954, A059889, A074249, A218358, A366620, A366633, A366634, A366635.

Cf. A046800, A133801, A366604, A366611, A366620, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(7^n - 1), ", "))

a(n) = omega(7^n-1) = A001221(A024075(n)).

A366633 Number of divisors of 7^n-1.

Original entry on oeis.org

4, 10, 12, 36, 8, 60, 16, 84, 64, 80, 16, 864, 8, 160, 96, 384, 16, 640, 16, 1536, 96, 160, 32, 16128, 32, 80, 1280, 1152, 32, 3840, 32, 1728, 384, 80, 128, 18432, 32, 160, 192, 14336, 32, 7680, 16, 4608, 2048, 160, 16, 147456, 256, 640, 768, 1152, 32, 25600

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(5)=8 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 168061}.

Max Alekseyev, Table of n, a(n) for n = 1..388

Cf. A024075, A000005, A057954, A059889, A074249, A218358, A366632, A366634, A366635.

Cf. A046801, A366575, A366602, A366612, A366621, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](7^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 7^Range[100]-1]
```
PARI
```
a(n) = numdiv(7^n-1);
```

a(n) = sigma0(7^n-1) = A000005(A024075(n)).

A366634 Sum of the divisors of 7^n-1.

Original entry on oeis.org

12, 124, 780, 7812, 33624, 354640, 1704240, 18929096, 97036800, 800520192, 3958188480, 56928231360, 193778020824, 1830926384640, 11181115146240, 115997032277280, 465294239722800, 5175558387507200, 22852200371636160, 287850454432579584, 1318081737957660000

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(5)=33624 because 7^5-1 has divisors {1, 2, 3, 6, 2801, 5602, 8403, 16806}.

Max Alekseyev, Table of n, a(n) for n = 1..388

Cf. A024075, A000203, A057954, A059887, A074249, A218358, A366632, A366633, A366635.

Cf. A075708, A366576, A366603, A366613, A366622, A366653, A366662, A102146, A366684, A366710.

a:=n->numtheory[sigma](7^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[1, 7^Range[30]-1]
```

a(n) = sigma(7^n-1) = A000203(A024075(n)).

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).

A381494 Smallest number with reciprocal of period length n in base 7.

Original entry on oeis.org

1, 2, 4, 9, 5, 2801, 36, 29, 64, 27, 11, 1123, 13, 16148168401, 113, 31, 17, 14009, 108, 419, 55, 261, 23, 47, 73, 2551, 53, 81, 145, 59, 99, 311, 256, 3631, 56036, 81229, 135, 223, 1676, 486643, 41, 83, 1017, 166003607842448777, 115, 837, 188, 13722816749522711, 153, 3529, 10204

Offset: 0

Erich Friedman, Feb 25 2025

For n > 1, a(n) is the smallest positive d such that d divides 7^n - 1 and does not divide any of 7^k - 1 for 0 < k < n.

			a(3)=9 since 1/9 has period 3 in base 7 (.053053053...) and no smaller number has this property.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Cf. A003060, A379640.

f:= proc(n) local d,k;
      for d in sort(convert(numtheory:-divisors(7^n-1),list)) do
        if andmap(k -> 7^k-1 mod d <> 0, [$1 .. n-1]) then return d fi
      od
end proc:
f(0):= 1: f(1):= 2:
map(f, [$0..80]); # Robert Israel, Feb 28 2025

a[n_] := First[Select[Divisors[7^n - 1], MultiplicativeOrder[7, #] == n &, 1]];
a[0] = 1; a[1] = 2; Table[a[n], {n, 0, 50}]

from sympy import divisors
def A381494(n): return next(d for d in divisors(7**n-1) if d>1 and all(pow(7,k,d)!=1 for k in range(1,n))) if n else 1 # Chai Wah Wu, Feb 28 2025

Conjecture: a(n) = A218358(n) for n>=2. - R. J. Mathar, Mar 03 2025

cp's OEIS Frontend

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

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

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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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A366632 Number of distinct prime divisors of 7^n - 1.

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A366633 Number of divisors of 7^n-1.

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A366634 Sum of the divisors of 7^n-1.

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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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