A337632 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, 3, 2, 1, 4, 2, 3, 1, 5, 2, 3, 2, 1, 6, 4, 6, 4, 6, 1, 7, 2, 3, 4, 3, 2, 1, 8, 2, 5, 2, 5, 2, 5, 1, 9, 2, 3, 4, 3, 2, 7, 2, 1, 10, 4, 6, 4, 10, 4, 6, 4, 10, 1, 11, 2, 3, 2, 3, 6, 3, 2, 3, 2, 1, 12, 4, 9, 4, 9, 4, 9, 4, 9, 4, 9, 1, 13, 2, 3, 4, 5, 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,  3, 2;
   4 | 1,  4, 2, 3;
   5 | 1,  5, 2, 3, 2;
   6 | 1,  6, 4, 6, 4,  6;
   7 | 1,  7, 2, 3, 4,  3, 2;
   8 | 1,  8, 2, 5, 2,  5, 2, 5;
   9 | 1,  9, 2, 3, 4,  3, 2, 7, 2;
  10 | 1, 10, 4, 6, 4, 10, 4, 6, 4, 10;
...
T(10,2) = 4 because
0^2 - 0 == 0 (mod 10),
1^2 - 1 == 0 (mod 10),
5^2 - 5 == 0 (mod 10), and
6^2 - 6 == 0 (mod 10).
		

Links

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

Crossrefs

Cf. A334006, A337633.

Programs

Haskell

a337632t 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

PARI

T(n,k) = sum(m=0, n-1, Mod(m,n)^k == Mod(m,n)); \\ Michel Marcus, Sep 13 2020

Formula

T(n,k) = A334006(n,k) * A337633(n,k).