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A070932
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Possible number of units in a finite (commutative or non-commutative) ring.
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3
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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, 40, 42, 44, 45, 46, 48, 49, 52, 54, 56, 58, 60, 62, 63, 64, 66, 70, 72, 78, 80, 81, 82, 84, 88, 90, 92, 93, 96, 98, 100
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OFFSET
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1,3
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COMMENTS
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This is a list of the numbers of units in R where R ranges over all finite commutative or non-commutative rings.
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.
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).
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
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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A282572 is the subsequence of odd terms.
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KEYWORD
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nonn,nice
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AUTHOR
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Sharon Sela (sharonsela(AT)hotmail.com), May 24 2002
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EXTENSIONS
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STATUS
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approved
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