A337632 - OEIS

%I #26 Sep 08 2022 08:46:25

%S 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,

%T 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,

%U 1,12,4,9,4,9,4,9,4,9,4,9,1,13,2,3,4,5,2

%N 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.

%H Peter Kagey, <a href="/A337632/b337632.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows, flattened)

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

%e Triangle begins:

%e   n\k| 0   1  2  3  4   5  6  7  8   9

%e   ---+--------------------------------

%e    1 | 1;

%e    2 | 1,  2;

%e    3 | 1,  3, 2;

%e    4 | 1,  4, 2, 3;

%e    5 | 1,  5, 2, 3, 2;

%e    6 | 1,  6, 4, 6, 4,  6;

%e    7 | 1,  7, 2, 3, 4,  3, 2;

%e    8 | 1,  8, 2, 5, 2,  5, 2, 5;

%e    9 | 1,  9, 2, 3, 4,  3, 2, 7, 2;

%e   10 | 1, 10, 4, 6, 4, 10, 4, 6, 4, 10;

%e ...

%e T(10,2) = 4 because

%e 0^2 - 0 == 0 (mod 10),

%e 1^2 - 1 == 0 (mod 10),

%e 5^2 - 5 == 0 (mod 10), and

%e 6^2 - 6 == 0 (mod 10).

%o (Haskell)

%o a337632t n k = length $ filter (\m -> (m^k - m) `mod` n == 0) [0..n-1]

%o (PARI) T(n,k) = sum(m=0, n-1, Mod(m,n)^k == Mod(m,n)); \\ _Michel Marcus_, Sep 13 2020

%o (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

%Y Cf. A334006, A337633.

%K nonn,tabl

%O 1,3

%A _Peter Kagey_, Sep 12 2020