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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).

%I A374001 #15 Jul 03 2024 07:46:17
%S A374001 4,16,32,96,160,256,480,704,896,1280,1152,1536,1920,3072,3744,4608,
%T A374001 3840,4224,5760,8640,7872,8448,9216,9600,9984,13824,16128,12288,14400,
%U A374001 20800,18432,25760,23040,23040,26240,38528,34176,42240,31104,48640,34560,48384,46080
%N 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).
%C A374001 Z_p[i] is a field iff p is a prime number congruent to 3 modulo 4.
%C A374001 a(n) is the number of generators of the multiplicative group Z_p[i] \ {0} (where p denotes A002145(n)).
%H A374001 Rémy Sigrist, <a href="/A374001/a374001.png">Scatterplot of (x, y) such that #{(x+i*y)^k, k >= 0} = p^2-1</a> (with p = A002145(62) = 647)
%H A374001 Rémy Sigrist, <a href="/A374001/a374001.txt">C++ program</a>
%H A374001 StackExchange, <a href="https://math.stackexchange.com/questions/1187066/z-pi-is-a-field">Z_p[i] is a field?</a>
%e A374001 For n = 2:
%e A374001 - the second prime number congruent to 3 modulo 4 is p = 7,
%e A374001 - the number of elements of {(x + i*y)^k, k >= 0} where x and y belong to Z_7 are:
%e A374001   x\y | 0   1   2   3   4   5   6
%e A374001   ----+--------------------------
%e A374001     0 | 2   4  12  12  12  12   4
%e A374001     1 | 1  24  48  48  48  48  24
%e A374001     2 | 3  48   8  16  16   8  48
%e A374001     3 | 6  48  16  24  24  16  48
%e A374001     4 | 3  48  16  24  24  16  48
%e A374001     5 | 6  48   8  16  16   8  48
%e A374001     6 | 2  24  48  48  48  48  24
%e A374001 - the number 48 appears 16 times, so a(2) = 16.
%o A374001 (C++) // See Links section.
%Y A374001 Cf. A002145, A271586, A373624.
%K A374001 nonn
%O A374001 1,1
%A A374001 _Rémy Sigrist_, Jun 24 2024