A334597 -id:A334597 - cp's OEIS frontend

A337820 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)).

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, 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, 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

Offset: 1

Juri-Stepan Gerasimov, Sep 23 2020

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)).

			The initial rows of the array are:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 1, 3, 3, 5, 3, 1, 3, 9, 1, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 5, 3, 3, 5, 3, 5, 3, 9, 5, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 1, 3, 7, 5, 7, 1, 3, 9, 1, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 5, 3, 3, 5, 3, 5, 3, 9, 5, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 1, 3, 3, 5, 3, 1, 3, 9, 1, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 3, 3, 5, 3, 7, 5, 7, 5, 3, 9, 5, ...
The initial antidiagonals are:
     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, 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, 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, 1, 7, 1, 5, 1, 3, 1, 1,
...

Columns 0-2: A000012, A026741, A000012.

Cf. A000010, A000012, A000027, A002322, A182816, A333570, A334006, A334597, A336664.

/* As triangle */ [[#[m: m in [0..n-k-1] | m^k mod (n-k) eq m]/
#[m: m in [0..n-k-1] | -m^k mod (n-k) eq m]: k in [0..n-1]]: n in [1..13]];

T(n, 2*k) = 1; 1 <= T(n, 2*k+1) <= n.

cp's OEIS Frontend

A337820 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)).

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