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).

1, 3, 8, 6, 24, 24, 48, 12, 24, 24, 120, 24, 168, 48, 24, 24, 288, 24, 360, 24, 48, 120, 528, 24, 120, 168, 72, 48, 840, 24, 960, 48, 120, 288, 48, 24, 1368, 360, 168, 24, 1680, 48, 1848, 120, 24, 528, 2208, 24, 336, 120, 288, 168, 2808, 72, 120, 48, 360, 840, 3480

Offset: 1

Author

Jianing Song, Mar 16 2019

Keywords

• nonn

Comments

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.

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).

a(n) is divisible by 24 unless n = 1, 2, 3, 4, 8.

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

Examples

			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.
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.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Jianing Song, Structure of (R/P^e)* for R the ring of integers of a quadratic field and P a prime ideal of R

Crossrefs

Cf. A007434.

Programs

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

Formula

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]