This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For n=8, there are 52 rings of order 8, 18 of which are noncommutative, so a(8)=18.
For n=4, there are 11 rings of order 4, 2 of which are without 1 and non-commutative, so a(4)= 2. Note that for these 2 rings, the abelian group under addition is the commutative Klein group Z/2Z + Z/2Z. These two rings are the last two rings described in the link _Greg Dresden_ in reference: Ring 22.NC.1 and Ring 22.NC.2.
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 *)
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
a(1) = 0 because the only ring with 1 element is the zero ring with the element 0, and for this ring, 0 and 1 coincide.
a(2) = 1, and for this corresponding ring with elements {0,a}, the multiplication that is defined by: 0*0 = 0*a = a*0 = a*a = 0 is commutative, also this ring is without unit, hence a(2) = 1; the Matrix ring {0,a} with coefficients from Z/2Z:
(0 0) (0 0)
0 = (0 0) a = (1 0) is such an example.
For n=8, there are 52 rings of order 8, 24 of which are commutative rings without 1, so a(8) = 24.
a(1) = 0 because the only ring with 1 element is the zero ring (see link) with the element 0, and for this ring, 0 and 1 coincide.
a(3) = 1 because the Matrix ring with 3 elements with coefficients from Z/3Z:
(0 0) (0 0) (0 0)
0 = (0 0), a = (1 0), b = (2 0)
is without 1 (note this ring is commutative) and there is no other such ring with 3 elements and without 1, hence a(3) = 1.
a(1) = 0 because there is a unique ring with 1 element, and a unique group of order 1, so 1 - 1 = 0.
The only near-ring of order 1 is the trivial ring, so a(1) = 1. There are 3 near-rings of order 2, all over Z2, so a(2) = 3. There are 5 near-rings of order 3, all over Z3, so a(3) = 5. There are 12 near-rings over Z4 and 23 near-rings over Z2^2, so a(4) = 12 + 23 = 35. There are 10 near-rings of order 5, all over Z5, so a(5) = 10. There are 60 near-rings over Z6 and 39 near-rings over S3, so a(6) = 60 + 39 = 99. There are 24 near-rings of order 7, all over Z7, so a(7) = 24. There are 135 near-rings over Z8, 1447 near-rings over D8, 281 near-rings over Q, 115 over Z4*Z2, and many over Z2^3, so a(8) > 1978.
a(15) = 1 + 2 + 2 + 55 + 2 + 4 + 2 + 801288 + 165 + 4 + 2 + 110 + 2 + 4 + 4 = 801647.
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