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
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
KEYWORD
nonn,base,fini,full
AUTHOR
EXTENSIONS
Offset corrected by Jon E. Schoenfield, Aug 12 2021
STATUS
approved