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A127708
Number of non-commutative 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
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 non-commuting elements.
LINKS
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 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.
CROSSREFS
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