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A345225
a(n) = order of 2-primary subgroup of the group K_n(Z).
2
1, 2, 2, 16, 1, 1, 1, 16, 1, 2, 2, 16, 1, 1, 1, 32, 1, 2, 2, 16, 1, 1, 1, 48, 1, 2, 2, 16, 1, 1, 1, 64, 1, 2, 2, 16, 1, 1, 1, 80, 1, 2, 2, 16, 1, 1, 1, 96, 1, 2, 2, 16, 1, 1, 1, 112, 1, 2, 2, 16, 1, 1, 1, 128, 1, 2, 2, 16, 1, 1, 1, 144, 1, 2, 2, 16, 1, 1, 1, 160
OFFSET
0,2
COMMENTS
The algebraic K-theory groups of the integers are not yet (as of June 2021) completely known, however the 2-primary part is settled.
In fact, the 2-primary part of the abelian group K_n(Z) is the cyclic group Z/a(n)Z.
REFERENCES
C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-9132-.
LINKS
C. Weibel, The 2-torsion in the K-theory of the integers, C. R. Acad. Sci. Paris 324 (1997), 615-620.
FORMULA
a(n) is:
1 if n mod 8 = 0, 4, 5, or 6
2 if n mod 8 = 1 or 2
16 if n mod 8 = 3
2*(n+1) if n mod 8 = 7.
(The main result of Weibel's 1997 paper on the 2-torsion in the K-theory of the integers; Corollary 9.8 of Weibel's K-book.)
PROG
(Python)
def a(n):
n_ = (n % 8)
d = {0:1, 1:2, 2:2, 3:16, 4:1, 5:1, 6:1}
if n_ == 7:
return 2*(n+1)
else:
return d[n_]
CROSSREFS
Sequence in context: A184718 A079897 A097540 * A373997 A112327 A152541
KEYWORD
easy,nonn
AUTHOR
Tom Harris, Jun 11 2021
STATUS
approved