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A345225
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a(n) = order of 2-primary subgroup of the group K_n(Z).
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2
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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
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OFFSET
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0,2
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COMMENTS
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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.
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REFERENCES
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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-.
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LINKS
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FORMULA
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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.)
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PROG
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(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_]
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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