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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 (list; graph; refs; listen; history; text; internal format)
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

Tom Harris, Table of n, a(n) for n = 0..9999

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 * A112327 A152541 A302339

Adjacent sequences:  A345222 A345223 A345224 * A345226 A345227 A345228

KEYWORD

easy,nonn

AUTHOR

Tom Harris, Jun 11 2021

STATUS

approved

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Last modified January 16 02:06 EST 2022. Contains 350374 sequences. (Running on oeis4.)