A070667 -id:A070667 - cp's OEIS frontend

A070680 Smallest m in range 1..phi(n) such that 11^m == 1 mod n, or 0 if no such number exists.

0, 1, 2, 2, 1, 2, 3, 2, 6, 1, 0, 2, 12, 3, 2, 4, 16, 6, 3, 2, 6, 0, 22, 2, 5, 12, 18, 6, 28, 2, 30, 8, 0, 16, 3, 6, 6, 3, 12, 2, 40, 6, 7, 0, 6, 22, 46, 4, 21, 5, 16, 12, 26, 18, 0, 6, 6, 28, 58, 2, 4, 30, 6, 16, 12, 0, 66, 16, 22, 3, 70, 6, 72, 6, 10, 6, 0, 12, 39, 4

Offset: 1

N. J. A. Sloane and Amarnath Murthy, May 08 2002

nonn

Cf. A070667-A070675, A002326, A070676, A053447, A070677, A070681, A070678, A053451, A070679, A070682, A070683.

[0] cat [Modorder(11, n): n in [2..100]]; // Vincenzo Librandi, Apr 01 2014

Table[SelectFirst[Range[EulerPhi[n]],PowerMod[11,#,n]==1&,0],{n,80}] (* Paul F. Marrero Romero, Oct 21 2024 *)

A070681 Smallest m in range 1..phi(2n+1) such that 6^m == 1 mod 2n+1, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 1, 2, 0, 10, 12, 0, 16, 9, 0, 11, 5, 0, 14, 6, 0, 2, 4, 0, 40, 3, 0, 23, 14, 0, 26, 10, 0, 58, 60, 0, 12, 33, 0, 35, 36, 0, 10, 78, 0, 82, 16, 0, 88, 12, 0, 9, 12, 0, 10, 102, 0, 106, 108, 0, 112, 11, 0, 16, 110, 0, 25, 126, 0, 130, 18, 0, 136, 23, 0, 60

Offset: 0

N. J. A. Sloane and Amarnath Murthy, May 08 2002

nonn

Cf. A070667-A070675, A002326, A070676, A053447, A070677, A070678, A053451, A070679, A070682, A070680, A070683.

A215653 a(n) = smallest positive m such that m^2 = 1+k*n with positive k.

Original entry on oeis.org

2, 3, 2, 3, 4, 5, 6, 3, 8, 9, 10, 5, 12, 13, 4, 7, 16, 17, 18, 9, 8, 21, 22, 5, 24, 25, 26, 13, 28, 11, 30, 15, 10, 33, 6, 17, 36, 37, 14, 9, 40, 13, 42, 21, 19, 45, 46, 7, 48, 49, 16, 25, 52, 53, 21, 13, 20, 57, 58, 11, 60, 61, 8, 31, 14, 23, 66, 33, 22, 29

Offset: 1

Zak Seidov, Aug 19 2012

nonn

Apparently a(n) = A070667(n) for n > 2. Note the linear patterns in the graph.

			a(1) = 2, k = 3; a(2) = 3, k = 4; a(3) = 2, k = 1; a(1000) = 249, k = 62.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
Dorin Andrica and Vlad Crişan, The Smallest Nontrivial Solution to x^k == 1 (mod n) and Related Sequences, Amer. Math. Monthly, Vol. 126, No. 2 (2019), pp. 173-178.

Cf. A061369, A070667, A076942.

Flatten[{2,Table[Select[Range[2,1000],PowerMod[#,2,k]==1&,1],{k,2,1000}]}] (* first 1000 terms *)

a(n) = {my(m = n + 1); while(!issquare(m), m += n); sqrtint(m);} \\ Amiram Eldar, Mar 16 2025

a(n) = sqrt(1+n*A076942(n)).

a(n) = sqrt(A061369(n)). - Amiram Eldar, Mar 16 2025

A228179 Irregular table where the n-th row consists of the square roots of 1 in Z_n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 3, 5, 7, 1, 8, 1, 9, 1, 10, 1, 5, 7, 11, 1, 12, 1, 13, 1, 4, 11, 14, 1, 7, 9, 15, 1, 16, 1, 17, 1, 18, 1, 9, 11, 19, 1, 8, 13, 20, 1, 21, 1, 22, 1, 5, 7, 11, 13, 17, 19, 23, 1, 24, 1, 25, 1, 26, 1, 13, 15, 27, 1, 28, 1, 11

Offset: 2

Tom Edgar, Aug 20 2013

Each 1 starts a new row.
This is a subsequence of A020652.
Row n has A060594(n) entries.
Each row forms a subgroup of the multiplicative group of units of Z_n.

			The table starts out as follows:
  1
  1 2
  1 3
  1 4
  1 5
  1 6
  1 3 5 7
  1 8
  1 9
  1 10
  1 5 7 11
  ...

Alois P. Heinz, Rows n = 2..2000 of irregular triangle, flattened

Cf. A060594, A038566, A020652, A082568.
Cf. A070667 (second column), A358016 (second-last column).
Cf. A277776 (nontrivial square roots of 1).

T:= n-> seq(`if`(k&^2 mod n=1, k, NULL), k=1..n-1):
seq(T(n), n=2..50);  # Alois P. Heinz, Aug 20 2013

Flatten[Table[Position[Mod[Range[n]^2, n], 1], {n, 2, 50}]] (* T. D. Noe, Aug 20 2013 *)

from itertools import chain, count, islice
from sympy.ntheory import sqrt_mod_iter
def A228179_gen(): # generator of terms
    return chain.from_iterable((sorted(sqrt_mod_iter(1,n)) for n in count(2)))
A228179_list = list(islice(A228179_gen(),30)) # Chai Wah Wu, Oct 26 2022

[[i for i in [1..k-1] if (i*i).mod(k)==1] for k in [2..n]] #changing n gives you the table up to the n-th row.

A091733 a(n) is the least m > 1 such that m^3 = 1 (mod n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 2, 9, 4, 11, 12, 13, 3, 9, 16, 17, 18, 7, 7, 21, 4, 23, 24, 25, 26, 3, 10, 9, 30, 31, 5, 33, 34, 35, 11, 13, 10, 7, 16, 41, 42, 25, 6, 45, 16, 47, 48, 49, 18, 51, 52, 9, 54, 19, 56, 9, 7, 59, 60, 61, 13, 5, 4, 65, 16, 67, 29, 69, 70, 11, 72, 25, 8, 47, 76, 45, 23, 55, 23

Offset: 1

David Wasserman, Mar 05 2004

a(n) <= n + 1; the inequality is strict iff n is divisible by 9 or by a prime congruent to 1 mod 3. - Robert Israel, May 27 2014

			a(7) = 2 because 2^3 is congruent to 1 (mod 7).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A070667, A076947, A083501.

m = 2; while mod(m^3 - 1, n); m = m + 1; end; m

A:= n -> min(select(t -> type((t^3-1)/n, integer), [$2 .. n+1]));
map(A, [$1 .. 1000]); # Robert Israel, May 27 2014

f[n_] := Block[{x = 2}, While[Mod[x^3 - 1, n] != 0, x++]; x]; Array[f, 79] (* Robert G. Wilson v, Mar 29 2016 *)

a(n) = my(k = 2); while(Mod(k, n)^3 != 1, k++); k; \\ Michel Marcus, Mar 30 2016

A070668 Smallest m in range 2..n-1 such that m^3 == 1 mod n, or 1 if no such number exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 3, 9, 1, 1, 1, 7, 7, 1, 4, 1, 1, 1, 1, 3, 10, 9, 1, 1, 5, 1, 1, 1, 11, 13, 10, 7, 16, 1, 1, 25, 6, 1, 16, 1, 1, 1, 18, 1, 1, 9, 1, 19, 1, 9, 7, 1, 1, 1, 13, 5, 4, 1, 16, 1, 29, 1, 1, 11, 1, 25, 8, 47, 1, 45, 23, 55, 23, 1, 28, 1, 1, 25

Offset: 1

N. J. A. Sloane, May 08 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A070667.

a:= proc(n) local m;
      for m from 2 to n-1 do
        if m &^ 3 mod n = 1 then return m fi
      od; 1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 29 2014

a[n_] := (For[m = 2, m <= n-1, m++, If[PowerMod[m, 3, n] == 1, Return[m]]]; 1); Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 20 2015 *)

a(n) = {for (m=2, n-1, if (lift(Mod(m, n)^3) == 1, return (m));); return (1);} \\ Michel Marcus, Jun 29 2014

A070669 Smallest m in range 2..n-1 such that m^4 == 1 mod n, or 1 if no such number exists.

Original entry on oeis.org

1, 1, 2, 3, 2, 5, 6, 3, 8, 3, 10, 5, 5, 13, 2, 3, 4, 17, 18, 3, 8, 21, 22, 5, 7, 5, 26, 13, 12, 7, 30, 7, 10, 13, 6, 17, 6, 37, 5, 3, 9, 13, 42, 21, 8, 45, 46, 5, 48, 7, 4, 5, 23, 53, 12, 13, 20, 17, 58, 7, 11, 61, 8, 15, 8, 23, 66, 13, 22, 13, 70, 17, 27, 31, 7

Offset: 1

N. J. A. Sloane, May 08 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A070667.

a:= proc(n) local m;
      for m from 2 to n-1 do
        if m &^ 4 mod n = 1 then return m fi
      od; 1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 29 2014

a[n_] := (For[m = 2, m <= n - 1, m++, If[PowerMod[m, 4, n] == 1, Return[m]]]; 1); Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 20 2015 *)

a(n) = {for (m=2, n-1, if (lift(Mod(m, n)^4) == 1, return (m));); return (1);} \\ Michel Marcus, Jun 29 2014

A070670 Smallest m in range 2..n-1 such that m^5 == 1 mod n, or 1 if no such number exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 5, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 9, 33, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 16, 1, 15, 1, 1, 1, 1, 37, 1, 1, 1, 1, 1, 9, 1, 1

Offset: 1

N. J. A. Sloane, May 08 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A070667.

a:= proc(n) local m;
      for m from 2 to n-1 do
        if m &^ 5 mod n = 1 then return m fi
      od; 1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 29 2014

a[n_] := (For[m = 2, m <= n - 1, m++, If[PowerMod[m, 5, n] == 1, Return[m]]]; 1); Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 20 2015 *)

a(n) = {for (m=2, n-1, if (lift(Mod(m, n)^5) == 1, return (m));); return (1);} \\ Michel Marcus, Jun 29 2014

A070671 Smallest m in range 2..n-1 such that m^6 == 1 mod n, or 1 if no such number exists.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 2, 3, 2, 9, 10, 5, 3, 3, 4, 7, 16, 5, 7, 9, 2, 21, 22, 5, 24, 3, 8, 3, 28, 11, 5, 15, 10, 33, 4, 5, 10, 7, 4, 9, 40, 5, 6, 21, 4, 45, 46, 7, 18, 49, 16, 3, 52, 17, 21, 3, 7, 57, 58, 11, 13, 5, 2, 31, 4, 23, 29, 33, 22, 9, 70, 5, 8, 11, 26, 7, 10, 17

Offset: 1

N. J. A. Sloane, May 08 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A070667.

a:= proc(n) local m;
      for m from 2 to n-1 do
        if m &^ 6 mod n = 1 then return m fi
      od; 1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 29 2014

a[n_] := Module[{m}, For[m = 2, m <= n-1, m++, If[PowerMod[m, 6, n] == 1, Return[m]]]; 1];
Array[a, 100] (* Jean-François Alcover, Nov 17 2020 *)

a(n) = {for (m=2, n-1, if (lift(Mod(m, n)^6) == 1, return (m));); return (1);} \\ Michel Marcus, Jun 29 2014

A070672 Smallest m in range 2..n-1 such that m^7 == 1 mod n, or 1 if no such number exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 20, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 7, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, May 08 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A070667.

a:= proc(n) local m;
      for m from 2 to n-1 do
        if m &^ 7 mod n = 1 then return m fi
      od; 1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 29 2014

a[n_] := Module[{m}, For[m = 2, m <= n-1, m++, If[PowerMod[m, 7, n] == 1, Return[m]]]; 1];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 20 2024 *)

a(n) = {for (m=2, n-1, if (lift(Mod(m, n)^7) == 1, return (m));); return (1);} \\ Michel Marcus, Jun 29 2014

cp's OEIS Frontend

A070680 Smallest m in range 1..phi(n) such that 11^m == 1 mod n, or 0 if no such number exists.

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A070681 Smallest m in range 1..phi(2n+1) such that 6^m == 1 mod 2n+1, or 0 if no such number exists.

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A215653 a(n) = smallest positive m such that m^2 = 1+k*n with positive k.

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A228179 Irregular table where the n-th row consists of the square roots of 1 in Z_n.

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A091733 a(n) is the least m > 1 such that m^3 = 1 (mod n).

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A070668 Smallest m in range 2..n-1 such that m^3 == 1 mod n, or 1 if no such number exists.

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A070669 Smallest m in range 2..n-1 such that m^4 == 1 mod n, or 1 if no such number exists.

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Maple

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A070670 Smallest m in range 2..n-1 such that m^5 == 1 mod n, or 1 if no such number exists.

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Maple