A329152 -id:A329152 - cp's OEIS frontend

A333268 Indices of unique values in A329152.

32, 240, 512, 1320, 2040, 2472, 2760, 4096, 5536, 5640, 6072, 9336, 9376, 12032, 12840, 13568, 14176, 15576, 16096, 17880, 18744, 20040, 20760, 21248, 21480, 21912, 22920, 23640, 24064, 25944, 28248, 28384, 29256, 31264, 31272

Offset: 1

Torlach Rush, Mar 13 2020

nonn

a(n) == 0 (mod 8).
If a*b = k*a(n) + 1 then b - a == 0 (mod 8), 1 < a < b < a(n), 1 <= k < a(n).
gcd(k, a*b) = 1.

			32 is a term because A329152(32) = 6 and 6 occurs exactly once in A329152.
240 is a term because A329152(240) = 24 and 24 occurs exactly once in A329152.
512 is a term because A329152(512) = 126 and 126 occurs exactly once in A329152.

Cf. A329152.

Subset of A008590.

A343292 Number of distinct results produced when generating a graphical image of each row of the multiplication table modulo n.

Original entry on oeis.org

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

Sébastien Dumortier, Apr 12 2021

nonn

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.

			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).

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.

Cf. A000010, A018253, A060594, A329152.

{1}~Join~Array[# - (EulerPhi[#] - Sum[Boole[Mod[k^2, #] == 1], {k, #}])/2 &, 69, 2] (* Michael De Vlieger, Apr 13 2021 *)

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

\\ 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

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

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A333268 Indices of unique values in A329152.

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A343292 Number of distinct results produced when generating a graphical image of each row of the multiplication table modulo n.

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