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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.

%I A337633 #21 Sep 08 2022 08:46:25
%S A337633 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,
%T A337633 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,
%U A337633 4,1,4,1,4,1,4,1,4,1,1,1,2,3,4,1,2,7,2
%N 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.
%H A337633 Peter Kagey, <a href="/A337633/b337633.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows, flattened)
%F A337633 T(n,k) = A337632(n,k)/A334006(n,k).
%e A337633 Triangle begins:
%e A337633   n\k| 0  1  2  3  4  5  6  7  8  9
%e A337633   ---+-----------------------------
%e A337633    1 | 1;
%e A337633    2 | 1, 2;
%e A337633    3 | 1, 1, 2;
%e A337633    4 | 1, 2, 2, 1;
%e A337633    5 | 1, 1, 2, 3, 2;
%e A337633    6 | 1, 2, 4, 2, 4, 2;
%e A337633    7 | 1, 1, 2, 1, 4, 1, 2;
%e A337633    8 | 1, 2, 2, 1, 2, 1, 2, 1;
%e A337633    9 | 1, 1, 2, 1, 4, 1, 2, 1, 2;
%e A337633   10 | 1, 2, 4, 6, 4, 2, 4, 6, 4, 2;
%e A337633 ...
%e A337633 T(10, 2) = 4 because
%e A337633 0^2 + 0 == 0 (mod 10),
%e A337633 4^2 + 4 == 0 (mod 10),
%e A337633 5^2 + 5 == 0 (mod 10), and
%e A337633 9^2 + 9 == 0 (mod 10).
%o A337633 (Haskell)
%o A337633 a337633t n k = length $ filter (\m -> (m^k + m) `mod` n == 0) [0..n-1]
%o A337633 (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 A337633 Cf. A333570, A334006, A337632.
%K A337633 nonn,tabl
%O A337633 1,3
%A A337633 _Peter Kagey_, Sep 12 2020