

A127708


Number of noncommutative rings with 1 containing n elements.


11



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
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OFFSET

1,16


COMMENTS

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


LINKS

Table of n, a(n) for n=1..63.
C. Noebauer, Home page [Archived copy as of 2008 from web.archive.org]


FORMULA

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


EXAMPLE

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 noncommutative ring (with unit) has cardinality 8 and is unique up to isomorphism; it can be represented as the ring of uppertriangular matrices of size 2 over F_2.
A037291(32) = 208, A127707(32) = 109, hence a(32) = 208  109 = 99.


CROSSREFS

Cf. A027623, A037291, A127707, A209401, A342376.
Sequence in context: A165970 A285180 A081357 * A094896 A277118 A200065
Adjacent sequences: A127705 A127706 A127707 * A127709 A127710 A127711


KEYWORD

nonn,nice,more


AUTHOR

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


EXTENSIONS

a(32)a(63) from Bernard Schott, Apr 19 2022
a(54) corrected by Andrey Zabolotskiy, Feb 02 2023


STATUS

approved



