cp's OEIS Frontend

A328410 Smallest m such that (Z/mZ)* = C_2 X C_(2n), or 0 if no such m exists, where (Z/mZ)* is the multiplicative group of integers modulo m.

%I A328410 #10 Oct 18 2019 17:04:18
%S A328410 8,15,21,32,33,35,0,51,57,55,69,0,0,87,77,128,0,95,0,123,129,115,141,
%T A328410 119,0,159,324,0,177,143,0,256,161,0,213,219,0,0,237,187,249,203,0,
%U A328410 267,209,235,0,291,0,303,309,0,321,327,253,339,0,295,0,287,0,0,381,512,393,299,0
%N A328410 Smallest m such that (Z/mZ)* = C_2 X C_(2n), or 0 if no such m exists, where (Z/mZ)* is the multiplicative group of integers modulo m.
%C A328410 If (Z/mZ)* is isomorphic to C_2 X C_(2k) for some k, let x be any element in (Z/mZ)* such that the multiplicative order of x is 2k and that x != -1, then {-1, x} generates (Z/mZ)*. For example, (Z/15Z)* = {+-1, +-2, +-4, +-8}, (Z/21Z)* = {+-1, +-5, +-4, +-20, +-16, +-17}.
%H A328410 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n">Multiplicative group of integers modulo n</a>
%e A328410 The solutions to (Z/mZ)* = C_2 X C_6 are m = 21, 28, 36 and 42, the smallest of which is 21, so a(3) = 21.
%o A328410 (PARI) a(n) = my(r=4*n, N=floor(exp(Euler)*r*log(log(r^2))+2.5*r/log(log(r^2)))); for(k=r+1, N+1, if(eulerphi(k)==r && lcm(znstar(k)[2])==r/2, return(k)); if(k==N+1, return(0)))
%Y A328410 Cf. A062373, A328411 (largest m).
%K A328410 nonn
%O A328410 1,1
%A A328410 _Jianing Song_, Oct 14 2019