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

1,8

Comments

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)).

Links

Table of n, a(n) for n=1..97.

Formula

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

Example

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,
...

Prog

(Magma) /* 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]];

Crossrefs

Columns 0-2: A000012, A026741, A000012.

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

Sequence in context: A328323 A090751 A344759 * A322127 A282496 A253238

Adjacent sequences: A337817 A337818 A337819 * A337821 A337822 A337823

Keyword

nonn,tabl

Author

Juri-Stepan Gerasimov, Sep 23 2020

Status

approved