cp's OEIS Frontend

A306933 Let k be any quadratic field such that all prime factors of n are inert in k, O_k be the corresponding ring of integers and G(n) = (O_k/nO_k)* be the multiplicative group of integers in O_k modulo n; then a(n) is the exponent of G(n).

%I A306933 #25 Jun 21 2025 16:16:56
%S A306933 1,3,8,6,24,24,48,12,24,24,120,24,168,48,24,24,288,24,360,24,48,120,
%T A306933 528,24,120,168,72,48,840,24,960,48,120,288,48,24,1368,360,168,24,
%U A306933 1680,48,1848,120,24,528,2208,24,336,120,288,168,2808,72,120,48,360,840,3480
%N A306933 Let k be any quadratic field such that all prime factors of n are inert in k, O_k be the corresponding ring of integers and G(n) = (O_k/nO_k)* be the multiplicative group of integers in O_k modulo n; then a(n) is the exponent of G(n).
%C A306933 The exponent of a finite group G is the least positive integer k such that x^k = e for every x in G, where e is the group identity.
%C A306933 Let k be any quadratic field. If a prime p is inert in k, then G(p^e) is isomorphic to C_(p^(e-1)) X C_((p^2-1)*p^(e-1)) if p is an odd prime; G(2^e) is isomorphic to C_3 if e = 1 and C_2 X C_(2^(e-2)) X C(3*2^(e-1)) if e >= 2. By the Chinese Remainder Theorem, if n = Product_{i=1..m} (p_i)^(e_i), then G(n) = G((p_1)^(e_1)) X G((p_2)^(e_2)) X ... G((p_m)^(e_m)). The order of G(n) is A007434(n).
%C A306933 a(n) is divisible by 24 unless n = 1, 2, 3, 4, 8.
%C A306933 Equivalently, G(p^e) can be defined as (Z_{p^2}/p^eZ_{p^2})*, where Z_{p^2} is the ring of integers of the field Q_{p^2} (with a unique maximal ideal pZ_{p^2}), and Q_{p^2} is the unique unramified quadratic extension of the p-adic field Q_p. - _Jianing Song_, Jun 19 2025
%H A306933 Jianing Song, <a href="/A306933/b306933.txt">Table of n, a(n) for n = 1..10000</a>
%H A306933 Jianing Song, <a href="/A306933/a306933.pdf">Structure of (R/P^e)* for R the ring of integers of a quadratic field and P a prime ideal of R</a>
%F A306933 a(p^e) = (p^2-1)*p^(e-1); if n = Product_{i=1..m} (p_i)^(e_i), then a(n) = lcm(a((p_1)^(e_1)), a((p_2)^(e_2)), ..., a((p_m)^(e_m))). [Simplified by _Jianing Song_, Feb 02 2020]
%e A306933 Let n = 10 = 2 * 5 and k = Q(sqrt(-3)); then both 2 and 5 are inert in k. G(10) = (O_k/10O_k)* = (O_k/2O_k)* X (O_k/5O_k)* = C_3 X C_24, the exponent of which is 24, so a(10) = 24.
%e A306933 Let n = 45 = 3^2 * 5 and k = Q(sqrt(2)); then both 3 and 5 are inert in k. G(45) = (O_k/45O_k)* = (O_k/3^2*O_k)* X (O_k/5O_k)* = (C_3 X C_24) X C_24, the exponent of which is 24, so a(45) = 24.
%o A306933 (PARI) a(n) = my(r=1, f=factor(n)); for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]); r=lcm(r, (p^2-1)*p^(e-1))); r \\ Corrected by _Jianing Song_, Feb 02 2020
%Y A306933 Cf. A007434.
%K A306933 nonn
%O A306933 1,2
%A A306933 _Jianing Song_, Mar 16 2019