A218357 -id:A218357 - cp's OEIS frontend

A059887 a(n) = |{m : multiplicative order of 5 mod m=n}|.

3, 5, 3, 12, 9, 37, 3, 28, 18, 47, 3, 180, 3, 53, 81, 176, 9, 446, 21, 564, 39, 117, 9, 884, 180, 53, 360, 244, 21, 5959, 9, 800, 39, 111, 369, 9536, 21, 483, 39, 5476, 9, 18289, 9, 1140, 2958, 111, 3, 9424, 6, 3852, 177, 884, 21, 81048, 561, 1188, 69, 227, 9

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(5).

Also, number of primitive factors of 5^n - 1 (cf. A218357). - Max Alekseyev, May 03 2022

Max Alekseyev, Table of n, a(n) for n = 1..502

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), this sequence (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A001692, A053447, A057956, A058945, A074479, A143665, A212485, A218357
Column k=5 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(5^d-1), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 12 2012

a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 5^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 13 2024, after Alois P. Heinz *)

a(n) = sumdiv(n, d, moebius(n/d)*numdiv(5^d-1)); \\ Michel Marcus, Dec 13 2024

a(n) = Sum_{d|n} mu(n/d)*tau(5^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A143665 a(n) is the least prime such that the multiplicative order of 5 mod a(n) equals n.

Original entry on oeis.org

2, 3, 31, 13, 11, 7, 19531, 313, 19, 521, 12207031, 601, 305175781, 29, 181, 17, 409, 5167, 191, 41, 379, 23, 8971, 390001, 101, 5227, 109, 234750601, 59, 61, 1861, 2593, 199, 3061, 211, 37, 149, 761, 79, 241, 2238236249, 43, 1644512641, 89, 1171, 47

Offset: 1

Vladimir Shevelev, Aug 28 2008

nonn

Max Alekseyev, Table of n, a(n) for n = 1..520

Cf. A064081, A218357.

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).

p = 2; t = Table[0, {100}]; While[p < 3000000001, a = MultiplicativeOrder[5, p]; If[0 < a < 101 && t[[a]] == 0, t[[a]] = p]; p = NextPrime@ p]; t (* Robert G. Wilson v, Oct 13 2014 *)

a(23)-a(40) from Robert G. Wilson v, Oct 13 2014

a(41)-a(46) from Robert G. Wilson v, Oct 15 2014

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) = {}.

A366613 Sum of the divisors of 5^n-1.

Original entry on oeis.org

7, 60, 224, 1736, 6048, 49920, 136724, 1107792, 3718400, 27060480, 85449224, 869499904, 2136230474, 15820920000, 61359427584, 461863805760, 1338408456700, 13177159680000, 33558717136896, 301282248701952, 863701914880000, 6313641012910080, 20863951122979048

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=224 because 5^3-1 has divisors {1, 2, 4, 31, 62, 124}.

Max Alekseyev, Table of n, a(n) for n = 1..502

Cf. A024049, A000203, A057956, A059887, A074479, A143665, A218357, A295502, A366611, A366612.

Cf. A075708, A366576, A366603, A366622, A366634, A366653, A366662, A102146, A366684, A366710.

a:=n->numtheory[sigma](5^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[1, 5^Range[30]-1]
```

a(n) = sigma(5^n-1) = A000203(A024049(n)).

A366612 Number of divisors of 5^n-1.

Original entry on oeis.org

3, 8, 6, 20, 12, 48, 6, 48, 24, 64, 6, 240, 6, 64, 96, 224, 12, 512, 24, 640, 48, 128, 12, 1152, 192, 64, 384, 320, 24, 6144, 12, 1024, 48, 128, 384, 10240, 24, 512, 48, 6144, 12, 18432, 12, 1280, 3072, 128, 6, 10752, 12, 4096, 192, 960, 24, 81920, 576, 1536

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=6 because 5^3-1 has divisors {1, 2, 4, 31, 62, 124}.

Max Alekseyev, Table of n, a(n) for n = 1..502

Cf. A024049, A000005, A057956, A059887, A074479, A143665, A295502, A218357, A366611, A366613.

Cf. A046801, A366575, A366602, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](5^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 5^Range[100]-1]
```
PARI
```
a(n) = numdiv(5^n-1);
```

a(n) = sigma0(5^n-1) = A000005(A024049(n)).

A366611 Number of distinct prime divisors of 5^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 4, 4, 5, 2, 6, 2, 5, 6, 6, 3, 7, 4, 8, 5, 6, 3, 8, 7, 5, 8, 7, 4, 11, 3, 8, 5, 6, 8, 11, 4, 8, 5, 11, 3, 12, 3, 9, 11, 6, 2, 11, 3, 11, 7, 8, 4, 14, 8, 9, 6, 7, 3, 17, 4, 7, 10, 11, 7, 12, 6, 11, 8, 14, 3, 16, 4, 8, 15, 11, 6, 11, 4, 15

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502

Cf. A024049, A001221, A057956, A059887, A074479, A143665, A218357, A295502, A366612, A366613.

Cf. A046800, A133801, A366604, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(5^n - 1), ", "))

a(n) = omega(5^n-1) = A001221(A024049(n)).

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A059887 a(n) = |{m : multiplicative order of 5 mod m=n}|.

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A143665 a(n) is the least prime such that the multiplicative order of 5 mod a(n) equals 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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A366613 Sum of the divisors of 5^n-1.

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A366612 Number of divisors of 5^n-1.

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A366611 Number of distinct prime divisors of 5^n - 1.

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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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