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A345267
Conjectural order of the torsion subgroup of the group K_n(Z) (the algebraic K-theory groups of the integers).
0
1, 2, 2, 48, 1, 1, 1, 240, 1, 2, 2, 1008, 1, 1, 1, 480, 1, 2, 2, 528, 1, 1, 691, 65520, 1, 2, 2, 48, 1, 1, 3617, 16320, 1, 2, 87734, 57456, 1, 1, 174611, 13200, 1, 2, 155366, 1104, 1, 1, 236364091, 131040, 1, 2, 1315862, 48, 1, 1, 3392780147, 6960, 1, 2
OFFSET
0,2
COMMENTS
a(n) is known for n != 0 mod 4 and is related to the Bernoulli numbers via the Riemann zeta function. See Section VI.9 of Weibel's K-book.
K_0(Z) = Z is classical, so a(0) = 1. Rognes proved that K_4(Z) = 0 in 2000 so a(4) = 1.
Otherwise the value of a(4i) = 1 is conjectural. Kurihara observed that this follows from the Kummer-Vandiver conjecture (and in fact is equivalent with it). The Kummer-Vandiver conjecture has been verified for primes up to 163 million, from which it follows that a(4i) must be at least that large if it is not 1.
REFERENCES
Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-91322.
LINKS
M. Kurihara, Some remarks on conjectures about cyclotomic fields and K-groups of Z, Compositio Math. 81 (1992), 223-236.
J. Rognes, K_4(Z) is the trivial group, Topology 30 (2000), 267-281.
FORMULA
Let k be the integer part of 1 + n/4.
a(n) is:
1 if n = 0 mod 8
2 if n = 1 mod 8
2*A000367(k) if n = 2 mod 8
2*A006863(k) if n = 3 mod 8
1 if n = 4 mod 8
1 if n = 5 mod 8
-1*A000367(k) if n = 6 mod 8
A006863(k) if n = 7 mod 8.
PROG
(Python)
from sympy import bernoulli
def a(n):
n_ = n % 8
k = n//4 + 1
b = bernoulli(2*k)/(4*k)
d = {0:1, 1:2, 4:1, 5:1}
if n_ == 2:
return 2 * b.numerator()
elif n_ == 3:
return 2 * b.denominator()
elif n_ == 6:
return -1 * b.numerator()
elif n_ == 7:
return b.denominator()
else:
return d[n_]
CROSSREFS
Cf. A345225 (the order of the 2-primary subgroup, divides a(n)).
A000367 / A006863 (numerator / denominator of B_2n/4n, where B_m are the Bernoulli numbers).
Sequence in context: A015167 A086204 A318085 * A316090 A345463 A322106
KEYWORD
nonn,easy
AUTHOR
Tom Harris, Jun 12 2021
STATUS
approved