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).

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)).

Links

Table of n, a(n) for n=1..43.

Rémy Sigrist, Scatterplot of (x, y) such that #{(x+i*y)^k, k >= 0} = p^2-1 (with p = A002145(62) = 647)

Rémy Sigrist, C++ program

StackExchange, Z_p[i] is a field?

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

Cf. A002145, A271586, A373624.

Sequence in context: A126032 A296819 A034713 * A101653 A043100 A329853

Adjacent sequences: A373998 A373999 A374000 * A374002 A374003 A374004

Keyword

nonn

Author

Rémy Sigrist, Jun 24 2024

Status

approved