%I #56 Dec 24 2021 11:01:13
%S 0,1,2,3,4,6,7,8,9,10,12,14,15,16,18,20,21,22,24,26,27,28,30,31,32,36,
%T 40,42,44,45,46,48,49,52,54,56,58,60,62,63,64,66,70,72,78,80,81,82,84,
%U 88,90,92,93,96,98,100
%N Possible number of units in a finite (commutative or non-commutative) ring.
%C This is a list of the numbers of units in R where R ranges over all finite commutative or non-commutative rings.
%C By considering the ring Z_n and the finite fields GF(q) this sequence contains the values of the Euler function phi(n) (A000010) and prime powers - 1 (A181062). By taking direct product of rings, if n and m belong to the sequence then so does m*n.
%C Eric M. Rains has shown that these rules generate all terms of this sequence. More precisely, he shows this sequence (with 0 removed) is the multiplicative monoid generated by all numbers of the form q^n-q^{n-1} for n >= 1 and q a prime power (see Rains link).
%C Since the number of units of F_q[X]/(X^n) is q^n - q^(n-1), restricting to finite commutative rings gives the same sequence. A296241, which is a proper supersequence, allows the ring R to be infinite. - _Jianing Song_, Dec 24 2021
%H Charles R Greathouse IV, <a href="/A070932/b070932.txt">Table of n, a(n) for n = 1..10000</a>
%H E. M. Rains, <a href="/A070932/a070932.txt">Comments on A070932</a>
%t max = 100; A000010 = EulerPhi[ Range[2*max]] // Union // Select[#, # <= max &] &; A181062 = Select[ Range[max], Length[ FactorInteger[#]] == 1 &] - 1; FixedPoint[ Select[ Outer[ Times, #, # ] // Flatten // Union, # <= max &] &, Union[A000010, A181062] ] (* _Jean-François Alcover_, Sep 10 2013 *)
%o (PARI) list(lim)=my(P=1, q, v, u=List()); forprime(p=2, default(primelimit), if(eulerphi(P*=p)>=lim, q=p; break)); v=vecsort(vector(P/q*lim\eulerphi(P/q), k, eulerphi(k)), , 8); v=select(n->n<=lim, v); forprime(p=2, sqrtint(lim\1+1), P=p; while((P*=p) <= lim+1, listput(u, P-1))); v=vecsort(concat(v, Vec(u)), , 8); u=List([0]); while(#u, v=vecsort(concat(v, Vec(u)),,8); u=List(); for(i=3,#v, for(j=i,#v,P=v[i]*v[j]; if(P>lim,break); if(!vecsearch(v, P), listput(u, P))))); v \\ _Charles R Greathouse IV_, Jan 08 2013
%Y Cf. A000010, A002202, A000252, A000961, A181062, A221178.
%Y A000252 is a subsequence.
%Y A282572 is the subsequence of odd terms.
%Y Proper subsequence of A296241.
%Y The main entries concerned with the enumeration of rings are A027623, A037234, A037291, A037289, A038538, A186116.
%K nonn,nice
%O 1,3
%A Sharon Sela (sharonsela(AT)hotmail.com), May 24 2002
%E Entry revised by _N. J. A. Sloane_, Jan 06 2013, Jan 08 2013
%E Definition clarified by _Jianing Song_, Dec 24 2021