A337633 Triangle read by rows: T(n,k) is the number of nonnegative integers m < n such that m^k + m == 0 (mod n), where 0 <= k < n.

1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 3, 2, 1, 2, 4, 2, 4, 2, 1, 1, 2, 1, 4, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 4, 6, 4, 2, 4, 6, 4, 2, 1, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 1, 1, 2, 3, 4, 1, 2, 7, 2

Offset: 1

Author

Peter Kagey, Sep 12 2020

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  n\k| 0  1  2  3  4  5  6  7  8  9
  ---+-----------------------------
   1 | 1;
   2 | 1, 2;
   3 | 1, 1, 2;
   4 | 1, 2, 2, 1;
   5 | 1, 1, 2, 3, 2;
   6 | 1, 2, 4, 2, 4, 2;
   7 | 1, 1, 2, 1, 4, 1, 2;
   8 | 1, 2, 2, 1, 2, 1, 2, 1;
   9 | 1, 1, 2, 1, 4, 1, 2, 1, 2;
  10 | 1, 2, 4, 6, 4, 2, 4, 6, 4, 2;
...
T(10, 2) = 4 because
0^2 + 0 == 0 (mod 10),
4^2 + 4 == 0 (mod 10),
5^2 + 5 == 0 (mod 10), and
9^2 + 9 == 0 (mod 10).
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)

Crossrefs

Cf. A333570, A334006, A337632.

Programs

Haskell

a337633t n k = length $ filter (\m -> (m^k + m) `mod` n == 0) [0..n-1]

Magma

[[#[m: m in [0..n-1] | -m^k mod n eq m]: k in [0..n-1]]: n in [1..17]]; // Juri-Stepan Gerasimov, Oct 12 2020

Formula

T(n,k) = A337632(n,k)/A334006(n,k).