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A270120
Number of k with k^n=1 (mod n) and k^k=k (mod n); related to some groups of order n.
1
0, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 6, 1, 4, 1, 6, 3, 2, 1, 8, 5, 2, 3, 4, 1, 4, 1, 6, 1, 2, 1, 8, 1, 2, 3, 12, 1, 8, 1, 4, 3, 2, 1, 12, 7, 6, 1, 6, 1, 4, 5, 8, 3, 2, 1, 12, 1, 2, 9, 10, 1, 4, 1, 6, 1, 4, 1, 16, 1, 2, 5, 4, 1, 8, 1, 20, 9, 2, 1, 16, 1, 2, 1, 8, 1, 8, 1, 4, 3, 2, 1, 12, 1, 8, 3, 14
OFFSET
1,4
COMMENTS
Given integers n and k, consider the operation
o_k: Z_n x Z_n -> Z_n, (a, b) -> a + k^a * b (mod n).
(Z_n, o_k) is a group if k^n == 1 (mod n) and k^k == k (mod n).
The first condition is necessary to get the definition well-defined.
The second condition is necessary for the associative property.
a(n) gives the number of different k out of {1, 2, ..., n-1} that comply the conditions.
E.g., for n = 4, k = -1 (or, what is the same, k = 3) results in the Klein four-group. (a o_3 b := a + (-1)^a * b (mod 4).)
Note that different k can result in groups that are isomorphic to each other.
The neutral element is always 0.
The inverse element to a is always -a*k^(-a) (mod n).
EXAMPLE
a(4) = 2, because in Z_4, k == 1 and k == 3 are the only number out of {0, 1, 2, 3} with conditions k^k==k mod n and k^n==1 mod n.
a(8) = 4, because k can be out of {1, 3, 5, 7}.
a(18) = 4, because k can be out of {1, 7, 13, 17}.
If n is even, k == -1 (or, equivalently, k == n-1) is always to be counted. This group is isomorphic to the Dihedral group D_(n/2), with generating elements -1 and 2.
The following table shows the first results with n, k and the name of the group (due to A. D. Thomas and G. V. Wood: 'Group Tables', found by comparing the element-orders).
Note that for n=8, k=1 and k=5 result in Z8. None of the k results in Z2 x Z4 or in Z2 x Z2 x Z2.
Note that for n=9 all k are isomorphic to Z9, none to Z3 x Z3.
n=2, k=1: Z2
n=3, k=1: Z3
n=4, k=1: Z4
n=4, k=3: Z2 x Z2
n=5, k=1: Z5
n=6, k=1: Z6
n=6, k=5: D3
n=7, k=1: Z7
n=8, k=1: Z8
n=8, k=3: Q4
n=8, k=5: Z8
n=8, k=7: D4
n=9, k=1: Z9
n=9, k=4: Z9
n=9, k=7: Z9
n=10, k=1: Z10
n=10, k=9: D5
...
MATHEMATICA
Table[Length[ Select[Range[1, n-1], ((GCD[n, # - 1] > 1) && (PowerMod[#, n, n] == 1) && (PowerMod[#, # - 1, n] == 1)) &]], {n, 1, 100}]
PROG
(PARI) a(n) = sum(k=1, n-1, (Mod(k, n)^n == 1) && (Mod(k, n)^k == k)); \\ Michel Marcus, Mar 12 2016
CROSSREFS
Cf. A000001 (number of groups of order n).
Sequence in context: A285707 A164879 A200219 * A325567 A009195 A072994
KEYWORD
nonn,easy
AUTHOR
STATUS
approved