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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.

%I A337632 #26 Sep 08 2022 08:46:25
%S A337632 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 A337632 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 A337632 1,12,4,9,4,9,4,9,4,9,4,9,1,13,2,3,4,5,2
%N 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.
%H A337632 Peter Kagey, <a href="/A337632/b337632.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows, flattened)
%F A337632 T(n,k) = A334006(n,k) * A337633(n,k).
%e A337632 Triangle begins:
%e A337632   n\k| 0   1  2  3  4   5  6  7  8   9
%e A337632   ---+--------------------------------
%e A337632    1 | 1;
%e A337632    2 | 1,  2;
%e A337632    3 | 1,  3, 2;
%e A337632    4 | 1,  4, 2, 3;
%e A337632    5 | 1,  5, 2, 3, 2;
%e A337632    6 | 1,  6, 4, 6, 4,  6;
%e A337632    7 | 1,  7, 2, 3, 4,  3, 2;
%e A337632    8 | 1,  8, 2, 5, 2,  5, 2, 5;
%e A337632    9 | 1,  9, 2, 3, 4,  3, 2, 7, 2;
%e A337632   10 | 1, 10, 4, 6, 4, 10, 4, 6, 4, 10;
%e A337632 ...
%e A337632 T(10,2) = 4 because
%e A337632 0^2 - 0 == 0 (mod 10),
%e A337632 1^2 - 1 == 0 (mod 10),
%e A337632 5^2 - 5 == 0 (mod 10), and
%e A337632 6^2 - 6 == 0 (mod 10).
%o A337632 (Haskell)
%o A337632 a337632t n k = length $ filter (\m -> (m^k - m) `mod` n == 0) [0..n-1]
%o A337632 (PARI) T(n,k) = sum(m=0, n-1, Mod(m,n)^k == Mod(m,n)); \\ _Michel Marcus_, Sep 13 2020
%o A337632 (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 A337632 Cf. A334006, A337633.
%K A337632 nonn,tabl
%O A337632 1,3
%A A337632 _Peter Kagey_, Sep 12 2020