A127708 Number of non-commutative rings with 1 containing n elements.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 99, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Hugues Randriam (randriam(AT)enst.fr), Jan 24 2007, Jan 29 2007

Keywords

• nonn
• nice
• more

Comments

We consider rings in which multiplication is associative and has a unit, but where there is at least one pair of non-commuting elements.

Examples

			a(n)=0 for n<=7 and a(8)=1, so all rings (with unit) of cardinality at most 7 are commutative, while the smallest non-commutative ring (with unit) has cardinality 8 and is unique up to isomorphism; it can be represented as the ring of upper-triangular matrices of size 2 over F_2.
A037291(32) = 208, A127707(32) = 109, hence a(32) = 208 - 109 = 99.
		

Links

• C. Noebauer, Home page [Archived copy as of 2008 from web.archive.org]

Crossrefs

Cf. A027623, A037291, A127707, A209401, A342376.

Formula

a(n) = A037291(n) - A127707(n). - Bernard Schott, Apr 19 2022

Extensions

a(32)-a(63) from Bernard Schott, Apr 19 2022

a(54) corrected by Andrey Zabolotskiy, Feb 02 2023