A027623 is multiplicative?

[Christian G.Bower]

I dealt with some of the ring sequences in Mitch's first batch, but the
problem keeps coming back.

> Ok, in

> http://www.algebra.uni-linz.ac.at/~noebsi/pub/smallrings.ps

> I find

> Theorem 1. [Sho38] Every finite ring R can be uniquely (up to
> isomorphism) decomposed into a direct sum of rings of prime power order.

The Noebauer paper should settle that A027623  and A037234

thus, the ring sequence (as well as commutative, with unit, etc) is
multiplicative.

This covers A027623 and A037234.

I strongly suspect this is also true of A037292 (Non associative rings)
but I haven't had time to do the algebra myself to confirm it and obviously
the theorem does not address it.

Regarding Dave's observation:

> In A027623 (number of rings with n elements), I noticed that for p
> prime, a(p) = 2 and a(p^2) = 11 seem to be true.  However, for p^3,
> we have a(8) = 52 while a(27) = 59, so a(p^3) depends on p.  Then I
> found this enticing tidbit:

>   The paper by Antipkin/Elizarov also gives the number a(p^3) of rings
> of order p^3. - Hans H. Storrer (storrer(AT)math.unizh.ch), Sep 16 2003
...
> If A027623(p^3) were in the OEIS, it would start 52,59,...  But no such 
> sequence exists in the OEIS.

The Noebauer paper also states that

> THEOREM 4. [FW74,LW80] There are  (3p+4)n-4p+11  non-isomorphic rings 
> with additive group  Z/p^n + Z/p  (n>=2) if p is odd and 9n+2 if p=2. 

> THEOREM 5. [FW74,LW80] There are  p+27  non-isomorphic rings with 
> additive group  Z/p + Z/p + Z/p  if p is odd and 28 if p=2. 

> These two theorems give a total of 3p+50 non-isomorphic rings of order
> p^3 for odd p and 52 for p=2.  This result is also obtained by [AE83]. 

That result is given in the A027623 entry.

Christian