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

%I A008831 #24 Aug 06 2024 07:11:56
%S A008831 0,1,4,2,9,5,11,3,8,10,7,6
%N A008831 Discrete logarithm of n to the base 2 modulo 13.
%C A008831 This is also a (12,1)-sequence.
%C A008831 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.
%D A008831 I. M. Vinogradov, Elements of Number Theory, p. 220.
%H A008831 H. Y. Song and S. W. Golomb, <a href="https://citeseerx.ist.psu.edu/pdf/20e087f1e3d60c14180d4c27a3183f942de29e94">Generalized Welch-Costas sequences and their application to Vatican arrays</a>, in Proc. 2nd International Conference on Finite Fields: Theory, Algorithms and Applications (Las Vegas 1993) Contemp. Math. vol. 168 344 1994.
%H A008831 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiscreteLogarithm.html">Discrete Logarithm</a>.
%F A008831 2^a(n) == n (mod 13). - _Michael S. Branicky_, Aug 22 2021
%e A008831 From _Jon E. Schoenfield_, Aug 21 2021: (Start)
%e A008831 Sequence is a permutation of the 12 integers 0..11:
%e A008831    k     2^k  2^k mod 13
%e A008831   --  ------  ----------
%e A008831    0       1           1  so a(1)  =  0
%e A008831    1       2           2  so a(2)  =  1
%e A008831    2       4           4  so a(4)  =  2
%e A008831    3       8           8  so a(8)  =  3
%e A008831    4      16           3  so a(3)  =  4
%e A008831    5      32           6  so a(6)  =  5
%e A008831    6      64          12  so a(12) =  6
%e A008831    7     128          11  so a(11) =  7
%e A008831    8     256           9  so a(9)  =  8
%e A008831    9     512           5  so a(5)  =  9
%e A008831   10    1024          10  so a(10) = 10
%e A008831   11    2048           7  so a(7)  = 11
%e A008831   12    4096           1
%e A008831 but a(1) = 0, so the sequence is finite with 12 terms.
%e A008831 (End)
%p A008831 [ seq(numtheory[mlog](n, 2, 13), n=1..12) ];
%t A008831 a[1] = 0; a[n_] := MultiplicativeOrder[2, 13, {n}]; Array[a, 12] (* _Jean-François Alcover_, Feb 09 2018 *)
%o A008831 (Python)
%o A008831 from sympy.ntheory import discrete_log
%o A008831 def a(n): return discrete_log(13, n, 2)
%o A008831 print([a(n) for n in range(1, 13)]) # _Michael S. Branicky_, Aug 22 2021
%Y A008831 A row of A054503.
%K A008831 nonn,base,fini,full
%O A008831 1,3
%A A008831 _N. J. A. Sloane_