cp's OEIS Frontend

A319597 Number of conjugacy classes for a non-abelian group of order p^3, where p is prime: a(n) = p^2 + p - 1 where p = prime(n).

%I A319597 #25 Nov 07 2022 07:40:10
%S A319597 5,11,29,55,131,181,305,379,551,869,991,1405,1721,1891,2255,2861,3539,
%T A319597 3781,4555,5111,5401,6319,6971,8009,9505,10301,10711,11555,11989,
%U A319597 12881,16255,17291,18905,19459,22349,22951,24805,26731,28055,30101,32219,32941,36671
%N A319597 Number of conjugacy classes for a non-abelian group of order p^3, where p is prime: a(n) = p^2 + p - 1 where p = prime(n).
%C A319597 For a non-abelian group of order p^3, we can use the class equation, p-group has nontrivial center result, group modulo center is cyclic implies group is abelian result, and the orbit-stabilizer theorem to give the number of conjugacy classes and number of elements in each conjugacy class.
%C A319597 The elements of A028387 with prime index.
%H A319597 Robert Israel, <a href="/A319597/b319597.txt">Table of n, a(n) for n = 1..10000</a>
%F A319597 From _Amiram Eldar_, Nov 07 2022: (Start)
%F A319597 a(n) = A028387(A000040(n)-1).
%F A319597 Product_{n>=1} (1 + 1/a(n)) = A065489.
%F A319597 Product_{n>=1} (1 - 1/a(n)) = A065480. (End)
%e A319597 For p^3=2^3=8, the conjugacy classes of the Dihedral group = <r, s | r^4=1, s^2=1, srs=r^{-1}> are {1}, {r^2}, {r, r^3}, {s, sr^2}, {sr, sr^3}.
%p A319597 A028387:= n -> n^2+n-1:
%p A319597 seq(A028387(ithprime(i)),i=1..50); # _Robert Israel_, Dec 23 2018
%t A319597 f[n_]:=n^2 + n - 1 ; f[Prime[Range[43]]] (* _Amiram Eldar_, Nov 21 2018 *)
%o A319597 (PARI) a(n) = {my(p = prime(n)); p^2 + p - 1; } \\ _Amiram Eldar_, Nov 07 2022
%Y A319597 Cf. A000040, A028387, A065480, A065489.
%K A319597 nonn
%O A319597 1,1
%A A319597 _Juan Lanfranco_, Sep 23 2018