A008832 Discrete logarithm of n to the base 2 modulo 19.

0, 1, 13, 2, 16, 14, 6, 3, 8, 17, 12, 15, 5, 7, 11, 4, 10, 9

Offset

1,3

Comments

Equivalently, a(n) is the multiplicative order of n with respect to base 2 (modulo 19), i.e., a(n) is the base-2 logarithm of the smallest k such that 2^k mod 19 = n. - Jon E. Schoenfield, Aug 13 2021

References

Carl Friedrich Gauss, "Disquisitiones Arithmeticae", Yale University Press, 1965; see p. 37.

I. M. Vinogradov, Elements of Number Theory, p. 221.

Links

Table of n, a(n) for n=1..18.

Eric Weisstein's World of Mathematics, Discrete Logarithm.

Formula

2^a(n) == n (mod 19). - Michael S. Branicky, Aug 13 2021

Example

From Jon E. Schoenfield, Aug 13 2021: (Start)
Sequence is a permutation of the 18 integers 0..17:
   k     2^k  2^k mod 19
  --  ------  ----------
   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          16  so a(16) =  4
   5      32          13  so a(13) =  5
   6      64           7  so a(7)  =  6
   7     128          14  so a(14) =  7
   8     256           9  so a(9)  =  8
   9     512          18  so a(18) =  9
  10    1024          17  so a(17) = 10
  11    2048          15  so a(15) = 11
  12    4096          11  so a(11) = 12
  13    8192           3  so a(3)  = 13
  14   16384           6  so a(6)  = 14
  15   32768          12  so a(12) = 15
  16   65536           5  so a(5)  = 16
  17  131072          10  so a(10) = 17
  18  262144           1
but a(1) = 0, so the sequence is finite with 18 terms.
(End)

Maple

[ seq(mlog(n, 2, 19), n=1..18) ];

Mathematica

a[1]=0; a[n_]:=MultiplicativeOrder[2, 19, {n}]; Array[a, 18] (* Vincenzo Librandi, Mar 21 2020 *)

Prog

(PARI) a(n) = znlog(n, Mod(2, 19)); \\ Kevin Ryde, Aug 13 2021
(Python)
from sympy.ntheory import discrete_log
def a(n): return discrete_log(19, n, 2)
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Aug 13 2021

Crossrefs

Cf. A036120.

Sequence in context: A367936 A124686 A113807 * A180704 A040168 A176593

Adjacent sequences: A008829 A008830 A008831 * A008833 A008834 A008835

Keyword

nonn,base,fini,full

Author

N. J. A. Sloane

Extensions

Offset corrected by Jon E. Schoenfield, Aug 12 2021

Status

approved