cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A036173 Log base 2 (n) mod 61.

Original entry on oeis.org

0, 1, 6, 2, 22, 7, 49, 3, 12, 23, 15, 8, 40, 50, 28, 4, 47, 13, 26, 24, 55, 16, 57, 9, 44, 41, 18, 51, 35, 29, 59, 5, 21, 48, 11, 14, 39, 27, 46, 25, 54, 56, 43, 17, 34, 58, 20, 10, 38, 45, 53, 42, 33, 19, 37, 52, 32, 36, 31, 30
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • I. M. Vinogradov, Elements of Number Theory, p. 223.

Links

Crossrefs

Cf. A054503.

Programs

  • Maple
    a:= n-> numtheory[mlog](n, 2, 61):
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 21 2021
  • Mathematica
    a[1]=0; a[n_]:=MultiplicativeOrder[2, 61, {n}]; Array[a, 60] (* Vincenzo Librandi, Mar 12 2020 *)

Formula

Row n=18 of T(n,k) in A054503. - Georg Fischer, Jan 19 2020

Extensions

a(59)-a(60) from Georg Fischer, Jan 19 2020

A036175 Log base 7 (n) mod 71.

Original entry on oeis.org

0, 6, 26, 12, 28, 32, 1, 18, 52, 34, 31, 38, 39, 7, 54, 24, 49, 58, 16, 40, 27, 37, 15, 44, 56, 45, 8, 13, 68, 60, 11, 30, 57, 55, 29, 64, 20, 22, 65, 46, 25, 33, 48, 43, 10, 21, 9, 50, 2, 62, 5, 51, 23, 14, 59, 19, 42, 4, 3, 66, 69, 17, 53, 36, 67, 63, 47, 61, 41, 35
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • I. M. Vinogradov, Elements of Number Theory, p. 224

Links

Crossrefs

Cf. A054503.

Programs

  • Mathematica
    a[1]=0; a[n_]:=MultiplicativeOrder[7, 71, {n}]; Array[a, 70] (* Vincenzo Librandi, Apr 03 2020 *)

Formula

Row n=20 of T(n,k) in A054503. - Georg Fischer, Jan 19 2020

Extensions

a(60)-a(71) from Georg Fischer, Jan 19 2020

A036176 Log base 5 (n) mod 73.

Original entry on oeis.org

0, 8, 6, 16, 1, 14, 33, 24, 12, 9, 55, 22, 59, 41, 7, 32, 21, 20, 62, 17, 39, 63, 46, 30, 2, 67, 18, 49, 35, 15, 11, 40, 61, 29, 34, 28, 64, 70, 65, 25, 4, 47, 51, 71, 13, 54, 31, 38, 66, 10, 27, 3, 53, 26, 56, 57, 68, 43, 5, 23, 58, 19, 45, 48, 60, 69, 50, 37, 52, 42, 44, 36
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • I. M. Vinogradov, Elements of Number Theory, p. 224.

Links

Crossrefs

Cf. A054503.

Programs

  • Mathematica
    a[1]=0; a[n_]:=MultiplicativeOrder[5, 73, {n}]; Array[a, 72] (* Vincenzo Librandi, Apr 04 2020 *)

Formula

Row n=21 of T(n,k) in A054503. - Georg Fischer, Jan 19 2020

Extensions

a(59)-a(72) from Georg Fischer, Jan 19 2020

A036177 Log base 3 (n) mod 79.

Original entry on oeis.org

0, 4, 1, 8, 62, 5, 53, 12, 2, 66, 68, 9, 34, 57, 63, 16, 21, 6, 32, 70, 54, 72, 26, 13, 46, 38, 3, 61, 11, 67, 56, 20, 69, 25, 37, 10, 19, 36, 35, 74, 75, 58, 49, 76, 64, 30, 59, 17, 28, 50, 22, 42, 77, 7, 52, 65, 33, 15, 31, 71, 45, 60, 55, 24, 18, 73, 48, 29, 27, 41, 51, 14, 44, 23, 47, 40, 43, 39
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • I. M. Vinogradov, Elements of Number Theory, p. 224.

Links

Crossrefs

Cf. A054503.

Programs

  • Mathematica
    a[1]=0; a[n_]:=MultiplicativeOrder[3, 79, {n}]; Array[a, 78] (* Vincenzo Librandi, Apr 04 2020 *)

Formula

Row n=22 of T(n,k) in A054503. - Georg Fischer, Jan 19 2020

Extensions

a(60)-a(78) from Georg Fischer, Jan 19 2020

A036178 Log base 2 (n) mod 83.

Original entry on oeis.org

0, 1, 72, 2, 27, 73, 8, 3, 62, 28, 24, 74, 77, 9, 17, 4, 56, 63, 47, 29, 80, 25, 60, 75, 54, 78, 52, 10, 12, 18, 38, 5, 14, 57, 35, 64, 20, 48, 67, 30, 40, 81, 71, 26, 7, 61, 23, 76, 16, 55, 46, 79, 59, 53, 51, 11, 37, 13, 34, 19, 66, 39, 70, 6, 22, 15, 45, 58, 50, 36, 33, 65, 69, 21, 44, 49, 32, 68, 43, 31, 42, 41
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • I. M. Vinogradov, Elements of Number Theory, p. 225.

Links

Crossrefs

Cf. A054503.

Programs

  • Maple
    a:= n-> numtheory[mlog](n, 2, 83):
seq(a(n), n=1..82);  # Alois P. Heinz, Aug 21 2021
  • Mathematica
    a[1]=0; a[n_]:=MultiplicativeOrder[2, 83, {n}]; Array[a, 82] (* Vincenzo Librandi, Apr 04 2020 *)

Formula

Row n=23 of T(n,k) in A054503. - Georg Fischer, Jan 19 2020

Extensions

a(60)-a(82) from Georg Fischer, Jan 19 2020

A036179 Log base 3 (n) mod 89.

Original entry on oeis.org

0, 16, 1, 32, 70, 17, 81, 48, 2, 86, 84, 33, 23, 9, 71, 64, 6, 18, 35, 14, 82, 12, 57, 49, 52, 39, 3, 25, 59, 87, 31, 80, 85, 22, 63, 34, 11, 51, 24, 30, 21, 10, 29, 28, 72, 73, 54, 65, 74, 68, 7, 55, 78, 19, 66, 41, 36, 75, 43, 15, 69, 47, 83, 8, 5, 13, 56, 38, 58, 79, 62, 50, 20, 27, 53, 67, 77, 40, 42, 46, 4, 37, 61, 26, 76, 45, 60, 44
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • I. M. Vinogradov, Elements of Number Theory, p. 225.

Links

Crossrefs

Cf. A054503.

Programs

  • Mathematica
    a[1]=0; a[n_]:=MultiplicativeOrder[3, 89, {n}]; Array[a, 88] (* Vincenzo Librandi, Mar 28 2020 *)

Formula

Row n=24 of T(n,k) in A054503. - Georg Fischer, Jan 19 2020

Extensions

a(59)-a(88) from Georg Fischer, Jan 19 2020

A036180 Log base 5 (n) mod 97.

Original entry on oeis.org

0, 34, 70, 68, 1, 8, 31, 6, 44, 35, 86, 42, 25, 65, 71, 40, 89, 78, 81, 69, 5, 24, 77, 76, 2, 59, 18, 3, 13, 9, 46, 74, 60, 27, 32, 16, 91, 19, 95, 7, 85, 39, 4, 58, 45, 15, 84, 14, 62, 36, 63, 93, 10, 52, 87, 37, 55, 47, 67, 43, 64, 80, 75, 12, 26, 94, 57, 61, 51, 66, 11, 50, 28, 29, 72, 53, 21, 33, 30, 41, 88, 23, 17, 73, 90, 38, 83, 92, 54, 79, 56, 49, 20, 22, 82, 48
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • I. M. Vinogradov, Elements of Number Theory, p. 225.

Links

Crossrefs

Cf. A054503.

Programs

  • Mathematica
    a[1]=0; a[n_]:=MultiplicativeOrder[5, 97, {n}]; Array[a, 96] (* Vincenzo Librandi, Mar 28 2020 *)

Formula

Row n=25 of T(n,k) in A054503. - Georg Fischer, Jan 19 2020

Extensions

a(60)-a(96) from Georg Fischer, Jan 19 2020

A207331 Array of indices (or logarithms) Modd p for odd numbers smaller than p relative to basis of smallest primitive root.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 0, 1, 3, 4, 2, 0, 2, 3, 1, 4, 5, 0, 1, 5, 3, 2, 7, 4, 6, 0, 1, 4, 6, 2, 3, 8, 5, 7, 0, 1, 9, 6, 2, 4, 5, 10, 8, 3, 7, 0, 1, 10, 8, 2, 5, 12, 11, 7, 13, 9, 4, 6, 3
Offset: 1

Views

Author

Wolfdieter Lang, Mar 27 2012

Keywords

Comments

For Modd n (not to be confused with mod n) see a comment on A203571.
The row lengths sequence for this array is 1 for row no. 1 and (p(n)-1)/2 with p(n):=A000040(n) (the primes).
For the definition of the index of a reduced number a mod n (but here we use Modd n) relative to a primitive root mod n, see, e.g., the Apostol reference, p. 213, and the tables on pp. 216-7. This mod n array is found under A054503 if the smallest primitive root mod n is taken as base. Because of its properties the index ind_b(a) is also called log_b(a), with the base b.
Here for Modd n, n>=2, primitive roots exist only for the values n with A206550(n)>0. There the smallest positive primitive roots, called here B(n) are also found. The allowed n values are shown in A206551. The indices Modd p(n), p(n):=A000040(n) (the primes) are called Ind_B(p(n))(a), with the odd numbers a smaller than p(n): 2*m-1=1,3,...,p(n)-2, for m=1,2,..., (p(n)-1)/2.
For odd p(n) the index Ind_B(p(n))(2*m-1) is defined as the unique value k from {0,1,...,(p(n)-3)/2}, such that B(p(n))^k = 2*m-1, with the base B(p(n)) the smallest positive primitive root Modd p(n).

Examples

			n, p(n)\m 1   2   3  4  5   6   7   8    9   10   11 12  13 14
    2m-1: 1   3   5  7  9  11  13  15   17   19   21 23  25 27
1,   2:   0
2,   3:   0
3,   5:   0   1
4,   7:   0   1   2
5,  11:   0   1   3  4  2
6,  13:   0   2   3  1  4   5
7,  17:   0   1   5  3  2   7   4   6
8,  19:   0   1   4  6  2   3   8   5    7
9,  23:   0   1   9  6  2   4   5  10    8    3   7
10, 29:   0   1  10  8  2   5  12  11    7   13   9   4   6  3
...
a(6,5) =4 because the base B(13) is here A206550(13)=7, and 7^4 = 2401, 2401 (Modd 13) := 2401 (mod 13) = 9 = 2*5-1.

References

  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.

Crossrefs

Cf. A054503 (mod n case).

Formula

a(n,m) = Ind_B(p(n))(2*m-1), m=1,2,..., (p(n)-1)/2, n>=1. See the comment section for the definition of Ind_B(a).
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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