|
%I #20 Sep 08 2022 08:46:25
%S 1,1,1,1,1,1,1,3,1,1,1,2,1,1,1,1,5,1,3,1,1,1,3,1,3,1,1,1,1,7,1,1,1,3,
%T 1,1,1,4,1,3,1,3,1,1,1,1,9,1,3,1,5,1,3,1,1,1,5,1,5,1,3,1,3,1,1,1,1,11,
%U 1,3,1,3,1,1,1,3,1,1,1,6,1,1,1,5,1,3,1,3,1,1,1,1,13,1,3,1,3
%N Array read by antidiagonals: T(n,k) (n >= 1, k >= 0) is the ratio (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)).
%C Array read by antidiagonals: T(n,k) (n >=1, k >= 0) is part of n of the form (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)).
%F T(n, 2*k) = 1; 1 <= T(n, 2*k+1) <= n.
%e The initial rows of the array are:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 3, 3, 1, 3, 3, 5, 3, 1, 3, 9, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 3, 3, 5, 3, 3, 5, 3, 5, 3, 9, 5, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 3, 3, 1, 3, 7, 5, 7, 1, 3, 9, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 3, 3, 5, 3, 3, 5, 3, 5, 3, 9, 5, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 3, 3, 1, 3, 3, 5, 3, 1, 3, 9, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 3, 3, 5, 3, 7, 5, 7, 5, 3, 9, 5, ...
%e The initial antidiagonals are:
%e 1,
%e 1, 1,
%e 1, 1, 1,
%e 1, 3, 1, 1,
%e 1, 2, 1, 1, 1,
%e 1, 5, 1, 3, 1, 1,
%e 1, 3, 1, 3, 1, 1, 1,
%e 1, 7, 1, 1, 1, 3, 1, 1,
%e 1, 4, 1, 3, 1, 3, 1, 1, 1,
%e 1, 9, 1, 3, 1, 5, 1, 3, 1, 1,
%e 1, 5, 1, 5, 1, 3, 1, 3, 1, 1, 1,
%e 1, 11, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1,
%e 1, 6, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1,
%e 1, 13, 1, 3, 1, 3, 1, 7, 1, 5, 1, 3, 1, 1,
%e ...
%o (Magma) /* As triangle */ [[#[m: m in [0..n-k-1] | m^k mod (n-k) eq m]/
%o #[m: m in [0..n-k-1] | -m^k mod (n-k) eq m]: k in [0..n-1]]: n in [1..13]];
%Y Columns 0-2: A000012, A026741, A000012.
%Y Cf. A000010, A000012, A000027, A002322, A182816, A333570, A334006, A334597, A336664.
%K nonn,tabl
%O 1,8
%A _Juri-Stepan Gerasimov_, Sep 23 2020
|
