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
Examples
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)
References
- Carl Friedrich Gauss, "Disquisitiones Arithmeticae", Yale University Press, 1965; see p. 37.
- I. M. Vinogradov, Elements of Number Theory, p. 221.
Links
- Eric Weisstein's World of Mathematics, Discrete Logarithm.
Crossrefs
Cf. A036120.
Programs
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Maple
[ seq(mlog(n,2,19), n=1..18) ];
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Mathematica
a[1]=0; a[n_]:=MultiplicativeOrder[2, 19, {n}]; Array[a, 18] (* Vincenzo Librandi, Mar 21 2020 *) -
PARI
a(n) = znlog(n, Mod(2, 19)); \\ Kevin Ryde, Aug 13 2021
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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
Formula
2^a(n) == n (mod 19). - Michael S. Branicky, Aug 13 2021
Extensions
Offset corrected by Jon E. Schoenfield, Aug 12 2021
Comments