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A343292
Number of distinct results produced when generating a graphical image of each row of the multiplication table modulo n.
1
1, 2, 3, 4, 4, 6, 5, 8, 7, 9, 7, 12, 8, 12, 13, 14, 10, 16, 11, 18, 17, 18, 13, 24, 16, 21, 19, 24, 16, 28, 17, 26, 25, 27, 25, 32, 20, 30, 29, 36, 22, 38, 23, 36, 35, 36, 25, 44, 29, 41, 37, 42, 28, 46, 37, 48, 41, 45, 31, 56, 32, 48, 47, 50, 43, 58, 35, 54, 49, 60
OFFSET
1,2
COMMENTS
The k-th row of the multiplication tables can be shown graphically by drawing a line for each i from i to k * i (mod n). The direction of the lines is not important.
LINKS
Michael De Vlieger, Scatterplot of (n, a(n)) for n=1..2^16.
Michael De Vlieger, Annotated scatterplot of (n, a(n)) for n=1..240, labeling a(n), with color function related to ratio (a(n)+(n+3)/2)/((n-3)/2), black for prime n. Red dashed line has slope 1. Blue dashed line = (a(n)+3)/2.
Michael De Vlieger, Annotated scatterplot of (n, (a(n)+(n+3)/2)/((n-3)/2)) for n=1..2^12, the highest values are labeled a(n).
Steve Phelps, Modular Times Table, GeoGebra.
FORMULA
a(n) = n - A329152(n) = n - (A000010(n) - A060594(n))/2. - Andrew Howroyd, Apr 12 2021
a(p) = (p + 3)/2 for p prime. - Michael De Vlieger, Apr 13 2021
EXAMPLE
Modulo 11, the 2 and 6 time tables, the 3 and 4 time tables, the 5 and 9 time tables, and the 7 and 8 time tables give the same pattern. So there are only 7 different time tables (0,1,2,3,5,7 and 10).
MATHEMATICA
{1}~Join~Array[# - (EulerPhi[#] - Sum[Boole[Mod[k^2, #] == 1], {k, #}])/2 &, 69, 2] (* Michael De Vlieger, Apr 13 2021 *)
PROG
(PARI)
G(n, r)={Set(vector(n, i, my(j=i*r%n); [min(i, j), max(i, j)]))}
a(n)={#Set(vector(n, k, concat(G(n, k-1))))} \\ Andrew Howroyd, Apr 12 2021
(PARI) \\ here b(n) is A060594(n).
b(n)={my(o=valuation(n, 2)); 2^(omega(n>>o)+max(min(o-1, 2), 0))}
a(n)={n - (eulerphi(n)-b(n))/2} \\ Andrew Howroyd, Apr 12 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved