A008831 Discrete logarithm of n to the base 2 modulo 13.
0, 1, 4, 2, 9, 5, 11, 3, 8, 10, 7, 6
Offset: 1
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
- H. Y. Song and S. W. Golomb, Generalized Welch-Costas sequences and their application to Vatican arrays, in Proc. 2nd International Conference on Finite Fields: Theory, Algorithms and Applications (Las Vegas 1993) Contemp. Math. vol. 168 344 1994.
- Eric Weisstein's World of Mathematics, Discrete Logarithm.
Crossrefs
A row of A054503.
Programs
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Maple
[ seq(numtheory[mlog](n, 2, 13), n=1..12) ];
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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
Comments