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A374001
a(n) is the number of elements z of Z_p[i] such that #{z^k, k >= 0} = p^2-1 (where p denotes A002145(n), the n-th prime number congruent to 3 modulo 4).
1
4, 16, 32, 96, 160, 256, 480, 704, 896, 1280, 1152, 1536, 1920, 3072, 3744, 4608, 3840, 4224, 5760, 8640, 7872, 8448, 9216, 9600, 9984, 13824, 16128, 12288, 14400, 20800, 18432, 25760, 23040, 23040, 26240, 38528, 34176, 42240, 31104, 48640, 34560, 48384, 46080
OFFSET
1,1
COMMENTS
Z_p[i] is a field iff p is a prime number congruent to 3 modulo 4.
a(n) is the number of generators of the multiplicative group Z_p[i] \ {0} (where p denotes A002145(n)).
EXAMPLE
For n = 2:
- the second prime number congruent to 3 modulo 4 is p = 7,
- the number of elements of {(x + i*y)^k, k >= 0} where x and y belong to Z_7 are:
x\y | 0 1 2 3 4 5 6
----+--------------------------
0 | 2 4 12 12 12 12 4
1 | 1 24 48 48 48 48 24
2 | 3 48 8 16 16 8 48
3 | 6 48 16 24 24 16 48
4 | 3 48 16 24 24 16 48
5 | 6 48 8 16 16 8 48
6 | 2 24 48 48 48 48 24
- the number 48 appears 16 times, so a(2) = 16.
PROG
(C++) // See Links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jun 24 2024
STATUS
approved