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.

Showing 1-10 of 18 results. Next

A054505 Log_b 2 where b is smallest primitive root (A001918) mod n-th prime.

Original entry on oeis.org

1, 1, 2, 1, 1, 14, 1, 2, 1, 24, 1, 26, 27, 18, 1, 1, 1, 1, 6, 8, 4, 1, 16, 34, 1, 44, 1, 57, 12, 72, 1, 10, 1, 1, 70, 141, 1, 40, 1, 1, 1, 44, 34, 1, 106, 1, 180, 1, 21, 72, 66, 190, 235, 48, 190, 1, 154, 147, 204, 159, 1, 93, 22, 274, 1, 121, 304, 1, 1, 164, 314, 292, 1, 1, 134, 1
Offset: 2

Views

Author

N. J. A. Sloane, Apr 09 2000

Keywords

Examples

			Smallest primitive root mod 7 is 3; 2 = 3^2 mod 7; 7 is 4th prime; so a(4) = 2.

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Table 10.2, pp. 216-217.

Links

Crossrefs

Cf. table in A054503.
Cf. A001918, A054506, A054507, A054508, A054509, A054510, A054511, A054512, A054513.

Programs

  • Mathematica
    a[n_] := Module[{p, b, lg = 1}, b = PrimitiveRoot[p = Prime[n]]; While[ PowerMod[b, lg, p] != 2 , lg++]; lg]; Array[a, 100, 2] (* Jean-François Alcover, Sep 03 2016 *)

Extensions

More terms from James Sellers, Apr 09 2000

A054513 Log_b 10 where b is smallest primitive root (A001918) mod n-th prime.

Original entry on oeis.org

5, 10, 3, 17, 3, 23, 14, 24, 8, 10, 19, 48, 7, 23, 16, 34, 9, 66, 28, 86, 35, 25, 45, 48, 25, 95, 33, 47, 85, 87, 105, 32, 142, 16, 41, 40, 139, 157, 94, 35, 90, 46, 133, 47, 12, 119, 5, 204, 88, 115, 103, 191, 209, 54, 148, 110, 110, 174, 94, 218, 1, 244, 27, 1, 278, 315
Offset: 5

Views

Author

N. J. A. Sloane, Apr 09 2000

Keywords

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Table 10.2, pp. 216-217.

Links

Crossrefs

Cf. table in A054503.
Cf. A001918, A054505, A054506, A054507, A054508, A054509, A054510, A054511, A054512.

Programs

  • Mathematica
    a[n_] := Module[{p, b, lg = 1}, b = PrimitiveRoot[p = Prime[n]]; While[ PowerMod[b, lg, p] != 10, lg++]; lg]; Array[a, 100, 5] (* Jean-François Alcover, Sep 03 2016 *)
  • PARI
    a(n)=znlog(10,znprimroot(prime(n))) \\ Charles R Greathouse IV, Oct 03 2011

Extensions

More terms from James Sellers, Apr 09 2000

A054506 Log_b 3 where b is smallest primitive root (A001918) mod n-th prime.

Original entry on oeis.org

3, 1, 8, 4, 1, 13, 16, 5, 1, 26, 15, 1, 20, 17, 50, 6, 39, 26, 6, 1, 72, 1, 70, 69, 39, 70, 52, 1, 1, 72, 1, 41, 87, 81, 82, 101, 94, 27, 108, 56, 116, 84, 181, 1, 43, 1, 46, 208, 1, 74, 182, 16, 1, 50, 109, 117, 188, 1, 1, 157, 81, 164, 56, 249, 1, 314, 152, 26, 1, 186, 75
Offset: 3

Views

Author

N. J. A. Sloane, Apr 09 2000

Keywords

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Table 10.2, pp. 216-217.

Links

Crossrefs

Cf. table in A054503.
Cf. A001918, A054505, A054507, A054508, A054509, A054510, A054511, A054512, A054513.

Programs

  • Mathematica
    a[n_] := Module[{p, b, lg = 1}, b = PrimitiveRoot[p = Prime[n]]; While[ PowerMod[b, lg, p] != 3 , lg++]; lg]; Array[a, 100, 3] (* Jean-François Alcover, Sep 03 2016 *)

Extensions

More terms from James Sellers, Apr 09 2000

A054507 Log_b 4 where b is smallest primitive root (A001918) mod n-th prime.

Original entry on oeis.org

2, 4, 2, 2, 12, 2, 4, 2, 18, 2, 12, 12, 36, 2, 2, 2, 2, 12, 16, 8, 2, 32, 68, 2, 88, 2, 6, 24, 18, 2, 20, 2, 2, 140, 126, 2, 80, 2, 2, 2, 88, 68, 2, 14, 2, 138, 2, 42, 144, 132, 140, 220, 96, 118, 2, 38, 18, 128, 36, 2, 186, 44, 236, 2, 242, 272, 2, 2, 328, 270, 218, 2, 2, 268, 2
Offset: 3

Views

Author

N. J. A. Sloane, Apr 09 2000

Keywords

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Table 10.2, pp. 216-217.

Links

Crossrefs

Cf. table in A054503.
Cf. A001918, A054505, A054506, A054508, A054509, A054510, A054511, A054512, A054513.

Programs

  • Mathematica
    a[n_] := Module[{p, b, lg = 1}, b = PrimitiveRoot[p = Prime[n]]; While[ PowerMod[b, lg, p] != 4, lg++]; lg]; Array[a, 100, 3] (* Jean-François Alcover, Sep 03 2016 *)

Extensions

More terms from James Sellers, Apr 09 2000

A054508 Log_b 5 where b is smallest primitive root (A001918) mod n-th prime.

Original entry on oeis.org

5, 4, 9, 5, 16, 1, 22, 20, 23, 22, 25, 1, 47, 6, 22, 15, 28, 1, 62, 27, 70, 1, 24, 1, 47, 76, 83, 87, 46, 75, 86, 104, 112, 1, 15, 1, 39, 138, 156, 50, 1, 89, 138, 132, 89, 11, 98, 165, 138, 138, 130, 55, 1, 208, 170, 1, 186, 233, 173, 1, 196, 39, 243, 236, 33, 277, 314
Offset: 4

Views

Author

N. J. A. Sloane, Apr 09 2000

Keywords

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Table 10.2, pp. 216-217.

Links

Crossrefs

Cf. table in A054503.
Cf. A001918, A054505, A054506, A054507, A054509, A054510, A054511, A054512, A054513.

Programs

  • Mathematica
    a[n_] := Module[{p, b, lg = 1}, b = PrimitiveRoot[p = Prime[n]]; While[ PowerMod[b, lg, p] != 5, lg++]; lg]; Array[a, 100, 4] (* Jean-François Alcover, Sep 03 2016 *)

Extensions

More terms from James Sellers, Apr 09 2000

A054509 Log_b 6 where b is smallest primitive root (A001918) mod n-th prime.

Original entry on oeis.org

3, 9, 5, 15, 14, 18, 6, 25, 27, 1, 28, 38, 18, 51, 7, 40, 32, 14, 5, 73, 17, 8, 70, 83, 71, 1, 13, 73, 73, 11, 42, 88, 1, 67, 102, 134, 28, 109, 57, 160, 118, 182, 107, 44, 181, 47, 1, 73, 140, 132, 1, 49, 240, 110, 1, 59, 205, 160, 158, 174, 186, 18, 250, 122, 282, 153
Offset: 4

Views

Author

N. J. A. Sloane, Apr 09 2000

Keywords

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Table 10.2, pp. 216-217.

Links

Crossrefs

Cf. table in A054503.
Cf. A001918, A054505, A054506, A054507, A054508, A054510, A054511, A054512, A054513.

Programs

  • Mathematica
    a[n_] := Module[{p, b, lg = 1}, b = PrimitiveRoot[p = Prime[n]]; While[ PowerMod[b, lg, p] != 6, lg++]; lg]; Array[a, 100, 4] (* Jean-François Alcover, Sep 03 2016 *)

Extensions

More terms from James Sellers, Apr 09 2000

A054510 Log_b 7 where b is smallest primitive root (A001918) mod n-th prime.

Original entry on oeis.org

7, 11, 11, 6, 19, 12, 28, 32, 39, 35, 32, 14, 18, 49, 23, 1, 33, 53, 8, 81, 31, 9, 4, 43, 40, 8, 115, 96, 42, 50, 142, 67, 147, 73, 118, 95, 171, 15, 171, 104, 146, 142, 139, 210, 154, 107, 222, 1, 1, 248, 85, 79, 19, 142, 22, 182, 278, 213, 116, 140, 123, 50, 81, 318
Offset: 5

Views

Author

N. J. A. Sloane, Apr 09 2000

Keywords

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Table 10.2, pp. 216-217.

Links

Crossrefs

Cf. table in A054503.
Cf. A001918, A054505, A054506, A054507, A054508, A054509, A054511, A054512, A054513.

Programs

  • Mathematica
    a[n_] := Module[{p, b, lg = 1}, b = PrimitiveRoot[p = Prime[n]]; While[ PowerMod[b, lg, p] != 7, lg++]; lg]; Array[a, 100, 5] (* Jean-François Alcover, Sep 03 2016 *)

Extensions

More terms from James Sellers, Apr 09 2000

A054511 Log_b 8 where b is smallest primitive root (A001918) mod n-th prime.

Original entry on oeis.org

3, 3, 10, 3, 6, 3, 12, 3, 38, 39, 8, 3, 3, 3, 3, 18, 24, 12, 3, 48, 6, 3, 30, 3, 63, 36, 90, 3, 30, 3, 3, 60, 111, 3, 120, 3, 3, 3, 132, 102, 3, 120, 3, 96, 3, 63, 216, 198, 90, 205, 144, 46, 3, 192, 165, 52, 195, 3, 279, 66, 198, 3, 33, 240, 3, 3, 140, 226, 144, 3, 3, 20, 3
Offset: 5

Views

Author

N. J. A. Sloane, Apr 09 2000

Keywords

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Table 10.2, pp. 216-217.

Links

Crossrefs

Cf. table in A054503.
Cf. A001918, A054505, A054506, A054507, A054508, A054509, A054510, A054512, A054513.

Programs

  • Mathematica
    a[n_] := Module[{p, b, lg = 1}, b = PrimitiveRoot[p = Prime[n]]; While[ PowerMod[b, lg, p] != 8, lg++]; lg]; Array[a, 100, 5] (* Jean-François Alcover, Sep 03 2016 *)

Extensions

More terms from James Sellers, Apr 09 2000

A054512 Log_b 9 where b is smallest primitive root (A001918) mod n-th prime.

Original entry on oeis.org

6, 8, 2, 8, 10, 10, 2, 16, 30, 2, 40, 34, 42, 12, 12, 52, 12, 2, 62, 2, 44, 38, 78, 34, 104, 2, 2, 14, 2, 82, 26, 12, 8, 40, 22, 54, 38, 112, 42, 168, 166, 2, 86, 2, 92, 188, 2, 148, 124, 32, 2, 100, 218, 234, 100, 2, 2, 22, 162, 18, 112, 182, 2, 292, 304, 52, 2, 14, 150, 104
Offset: 5

Views

Author

N. J. A. Sloane, Apr 09 2000

Keywords

References

  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Table 10.2, pp. 216-217.

Links

Crossrefs

Cf. table in A054503.
Cf. A001918, A054505, A054506, A054507, A054508, A054509, A054510, A054511, A054513.

Programs

  • Mathematica
    a[n_] := Module[{p, b, lg = 1}, b = PrimitiveRoot[p = Prime[n]]; While[ PowerMod[b, lg, p] != 9, lg++]; lg]; Array[a, 100, 5] (* Jean-François Alcover, Sep 03 2016 *)

Extensions

More terms from James Sellers, Apr 09 2000

A008831 Discrete logarithm of n to the base 2 modulo 13.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

This is also a (12,1)-sequence.
Equivalently, a(n) is the multiplicative order of n with respect to base 2 (modulo 13), i.e., a(n) is the base-2 logarithm of the smallest k such that 2^k mod 13 = n.

Examples

			From _Jon E. Schoenfield_, Aug 21 2021: (Start)
Sequence is a permutation of the 12 integers 0..11:
   k     2^k  2^k mod 13
  --  ------  ----------
   0       1           1  so a(1)  =  0
   1       2           2  so a(2)  =  1
   2       4           4  so a(4)  =  2
   3       8           8  so a(8)  =  3
   4      16           3  so a(3)  =  4
   5      32           6  so a(6)  =  5
   6      64          12  so a(12) =  6
   7     128          11  so a(11) =  7
   8     256           9  so a(9)  =  8
   9     512           5  so a(5)  =  9
  10    1024          10  so a(10) = 10
  11    2048           7  so a(7)  = 11
  12    4096           1
but a(1) = 0, so the sequence is finite with 12 terms.
(End)

References

  • I. M. Vinogradov, Elements of Number Theory, p. 220.

Links

Crossrefs

A row of A054503.

Programs

  • Maple
    [ seq(numtheory[mlog](n, 2, 13), n=1..12) ];
  • Mathematica
    a[1] = 0; a[n_] := MultiplicativeOrder[2, 13, {n}]; Array[a, 12] (* Jean-François Alcover, Feb 09 2018 *)
  • Python
    from sympy.ntheory import discrete_log
def a(n): return discrete_log(13, n, 2)
print([a(n) for n in range(1, 13)]) # Michael S. Branicky, Aug 22 2021

Formula

2^a(n) == n (mod 13). - Michael S. Branicky, Aug 22 2021
Showing 1-10 of 18 results. Next

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

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