cp's OEIS Frontend

A380578 Number of nonisomorphic groups appearing as the group of units of the ring Z/kZ for every k such that phi(k) = n.

%I A380578 #10 Apr 01 2025 23:08:55
%S A380578 1,1,0,2,0,1,0,2,0,1,0,2,0,0,0,3,0,1,0,2,0,1,0,2,0,0,0,1,0,1,0,4,0,0,
%T A380578 0,3,0,0,0,3,0,1,0,1,0,1,0,3,0,0,0,1,0,1,0,1,0,1,0,2,0,0,0,5,0,1,0,0,
%U A380578 0,1,0,5,0,0,0,0,0,1,0,3,0,1,0,1,0,0,0,3,0,0,0
%N A380578 Number of nonisomorphic groups appearing as the group of units of the ring Z/kZ for every k such that phi(k) = n.
%C A380578 Every group of units is abelian.
%F A380578 a(n) <= A014197(n).
%F A380578 a(n) = 0 for every n belonging to A007617.
%e A380578 a(4) = 2 because of the 4 distinct k such that phi(k) = 4 there are 2 nonisomorphic group of units Z/kZ*: C_4, and C_2 x C_2.
%e A380578 a(40) = 3 because of the 9 distinct k such that phi(k) = 40 there are 3 nonisomorphic group of units Z/kZ*: C_40, C_20 x C_2, and C_10 x C_2 x C_2.
%e A380578 a(41) = 0 because there are no k such that phi(k) = 41.
%o A380578 (PARI) groupcount(n) = b=[]; if(n==1, b=concat(b,2), forstep(k=floor(exp(Euler)*n*log(log(n^2))+2.5*n/log(log(n^2))), n, -1, if(eulerphi(k)==n, b=concat(b,k)); if(k==n, b=concat(b,0)))); Z=[]; if(istotient(n)==0, return(0), for(m=2, b[1], if(eulerphi(m)<>n, next, W=[]; U=[]; D=divisors(eulerphi(m)); lambda=lcm(znstar(m)[2]); for(k=1, m-1, if(gcd(k,m)==1, U=concat(U, k))); for(j=1, length(D), if(D[j]>lambda, break); S=[]; for(r=1, eulerphi(m), if(znorder(Mod(U[r], m))==D[j], S=concat(S, U[r]))); W=concat(W, length(S)))); Z=concat(Z,[W]); Z=Set(Z)); return(length(Z)))
%Y A380578 Cf. A000010, A000688, A005277, A007617, A014197, A049283, A057635.
%K A380578 nonn
%O A380578 1,4
%A A380578 _Miles Englezou_, Mar 26 2025